Datasets:

hoskinson-center
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proof-pile

	:: Free Magmas
	:: by Marco Riccardi

	environ

	vocabularies NUMBERS, FUNCT_1, ORDINAL1, RELAT_1, XBOOLE_0, TARSKI, AFINSQ_1,
	SUBSET_1, YELLOW_6, ZFMISC_1, CLASSES1, PARTFUN1, ALGSTR_0, BINOP_1,
	EQREL_1, MSUALG_6, STRUCT_0, GROUP_6, MSSUBFAM, FUNCT_2, SETFAM_1,
	REALSET1, CIRCUIT2, CARD_1, XXREAL_0, FINSEQ_1, ARYTM_1, CARD_3, ARYTM_3,
	NAT_1, XCMPLX_0, MCART_1, NAT_LAT, ALGSTR_4;
	notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, SUBSET_1, CARD_1, XCMPLX_0,
	XFAMILY, RELAT_1, FUNCT_1, ORDINAL1, NUMBERS, SETFAM_1, FUNCT_6, FUNCT_2,
	XXREAL_0, NAT_1, CLASSES1, FINSEQ_1, CARD_3, AFINSQ_1, NAT_D, YELLOW_6,
	BINOP_1, STRUCT_0, ALGSTR_0, RELSET_1, GROUP_6, MCART_1, NAT_LAT,
	PARTFUN1, REALSET1, EQREL_1, ALG_1, GROUP_2;
	constructors NAT_1, CLASSES1, AFINSQ_1, NAT_D, YELLOW_6, BINOP_1, RELSET_1,
	FACIRC_1, GROUP_6, NAT_LAT, REALSET1, EQREL_1, XTUPLE_0, XFAMILY;
	registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, FINSET_1,
	XXREAL_0, XREAL_0, NAT_1, CARD_1, FUNCT_2, INT_1, STRUCT_0, RELSET_1,
	NAT_LAT, REALSET1, EQREL_1, GROUP_2, XTUPLE_0;
	requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
	definitions TARSKI;
	equalities TARSKI, ALGSTR_0, ORDINAL1;
	expansions TARSKI;
	theorems TARSKI, XBOOLE_1, ORDINAL1, RELAT_1, FUNCT_1, FUNCT_2, AFINSQ_1,
	FINSEQ_1, XREAL_1, NAT_1, XXREAL_0, XREAL_0, ZFMISC_1, CLASSES1, CARD_3,
	YELLOW_6, BINOP_1, RELSET_1, MCART_1, INT_1, SUBSET_1, XBOOLE_0, NAT_LAT,
	FUNCT_5, GROUP_6, PARTFUN1, REALSET1, SETFAM_1, EQREL_1, OSALG_4,
	GROUP_2, FUNCT_6, XTUPLE_0;
	schemes ORDINAL1, NAT_1, CLASSES1, FINSEQ_1, BINOP_1, FUNCT_2, EQREL_1,
	RELAT_1, XFAMILY;

	begin :: Preliminaries

	registration
	let X be set;
	let f be sequence of X;
	let n be Nat;
	cluster f\|n -> Sequence-like;
	correctness
	proof
	per cases;
	suppose A1: X <> {};
	n c= NAT; then
	n c= dom f by A1,FUNCT_2:def 1; then
	dom(f\|n) is ordinal by RELAT_1:62;
	hence thesis by ORDINAL1:def 7;
	end;
	suppose X = {}; then
	f = {};
	hence thesis;
	end;
	end;
	end;

	definition
	let X,x be set;
	func IFXFinSequence(x,X) -> XFinSequence of X equals :Def1:
	x if x is XFinSequence of X
	otherwise <%>X;
	correctness;
	end;

	definition
	let X be non empty set;
	let f be Function of X^omega, X;
	let c be XFinSequence of X;
	redefine func f.c -> Element of X;
	correctness
	proof
	c in X^omega by AFINSQ_1:def 7;
	hence thesis by FUNCT_2:5;
	end;
	end;

	theorem Th1:
	for X,Y,Z being set st Y c= the_universe_of X & Z c= the_universe_of X
	holds [:Y,Z:] c= the_universe_of X
	proof
	let X,Y,Z be set;
	assume Y c= the_universe_of X; then
	A1: Y c= Tarski-Class the_transitive-closure_of X by YELLOW_6:def 1;
	assume Z c= the_universe_of X; then
	Z c= Tarski-Class the_transitive-closure_of X by YELLOW_6:def 1;
	then
	[:Y,Z:] c= Tarski-Class the_transitive-closure_of X by A1,CLASSES1:28;
	hence [:Y,Z:] c= the_universe_of X by YELLOW_6:def 1;
	end;

	scheme FuncRecursiveUniq { F(set) -> set, F1,F2() -> Function }:
	F1() = F2()
	provided
	A1: dom F1() = NAT & for n being Nat holds F1().n = F(F1()\|n) and
	A2: dom F2() = NAT & for n being Nat holds F2().n = F(F2()\|n)
	proof
	reconsider L1=F1() as Sequence by A1,ORDINAL1:def 7;
	reconsider L2=F2() as Sequence by A2,ORDINAL1:def 7;
	A3: dom L1 = NAT &
	for B being Ordinal,T1 being Sequence st B in NAT & T1 = L1\|B
	holds L1.B = F(T1) by A1;
	A4: dom L2 = NAT &
	for B being Ordinal,T2 being Sequence st B in NAT & T2 = L2\|B
	holds L2.B = F(T2) by A2;
	L1 = L2 from ORDINAL1:sch 3(A3,A4);
	hence thesis;
	end;

	scheme FuncRecursiveExist { F(set) -> set }:
	ex f being Function st dom f = NAT &
	for n being Nat holds f.n = F(f\|n)
	proof
	consider L being Sequence such that
	A1: dom L = NAT and
	A2: for B being Ordinal,L1 being Sequence st B in NAT & L1 = L\|B
	holds L.B = F(L1) from ORDINAL1:sch 4;
	take L;
	thus dom L = NAT by A1;
	let n be Nat;
	reconsider B=n as Ordinal;
	B in NAT by ORDINAL1:def 12; then
	L.B = F(L\|B) by A2;
	hence thesis;
	end;

	scheme FuncRecursiveUniqu2
	{ X() -> non empty set, F(XFinSequence of X()) -> Element of X(),
	F1,F2() -> sequence of X()}:
	F1() = F2()
	provided
	A1: for n being Nat holds F1().n = F(F1()\|n) and
	A2: for n being Nat holds F2().n = F(F2()\|n)
	proof
	deffunc FX(set) = F(IFXFinSequence($1,X()));
	reconsider f1=F1() as Function;
	reconsider f2=F2() as Function;
	A3: dom f1 = NAT & for n being Nat holds f1.n = FX(f1\|n)
	proof
	thus dom f1 = NAT by FUNCT_2:def 1;
	let n be Nat;
	thus f1.n = F(F1()\|n) by A1 .= FX(f1\|n) by Def1;
	end;
	A4: dom f2 = NAT & for n being Nat holds f2.n = FX(f2\|n)
	proof
	thus dom f2 = NAT by FUNCT_2:def 1;
	let n be Nat;
	thus f2.n = F(F2()\|n) by A2 .= FX(f2\|n) by Def1;
	end;
	f1 = f2 from FuncRecursiveUniq(A3,A4);
	hence thesis;
	end;

	scheme FuncRecursiveExist2
	{ X() -> non empty set, F(XFinSequence of X()) -> Element of X() }:
	ex f being sequence of X() st
	for n being Nat holds f.n = F(f\|n)
	proof
	deffunc FX(set) = F(IFXFinSequence($1,X()));
	consider f be Function such that
	A1: dom f = NAT and
	A2: for n being Nat holds f.n = FX(f\|n) from FuncRecursiveExist;
	for y being object st y in rng f holds y in X()
	proof
	let y be object;
	assume y in rng f; then
	consider x be object such that
	A3: x in dom f & y = f.x by FUNCT_1:def 3;
	reconsider x as Nat by A1,A3;
	y = F(IFXFinSequence(f\|x,X())) by A3,A2;
	hence y in X();
	end; then
	rng f c= X(); then
	reconsider f9=f as sequence of X() by A1,FUNCT_2:2;
	take f9;
	let n be Nat;
	f.n = F(IFXFinSequence(f9\|n,X())) by A2
	.= F(f9\|n) by Def1;
	hence thesis;
	end;

	definition
	let f,g be Function;
	pred f extends g means
	dom g c= dom f & f tolerates g;
	end;

	registration
	cluster empty for multMagma;
	existence
	proof
	take multMagma(# {}, the BinOp of {} #);
	thus thesis;
	end;
	end;

	begin :: Equivalence Relations and Relators
	:: Ch I ?1.6 Def.11 Algebra I Bourbaki

	definition
	let M be multMagma;
	let R be Equivalence_Relation of M;
	attr R is compatible means :Def3:
	for v,v9,w,w9 being Element of M st v in Class(R,v9) & w in Class(R,w9)
	holds vw in Class(R,v9w9);
	end;

	registration
	let M be multMagma;
	cluster nabla the carrier of M -> compatible;
	correctness
	proof
	set R = nabla the carrier of M;
	let v,v9,w,w9 be Element of M;
	assume v in Class(R,v9) & w in Class(R,w9);
	then
	A1: M is non empty;
	then Class(R,v9*w9) = the carrier of M by EQREL_1:26;
	hence thesis by A1,SUBSET_1:def 1;
	end;
	end;

	registration
	let M be multMagma;
	cluster compatible for Equivalence_Relation of M;
	correctness
	proof
	take nabla the carrier of M;
	thus thesis;
	end;
	end;

	theorem Th2:
	for M being multMagma, R being Equivalence_Relation of M holds
	R is compatible iff for v,v9,w,w9 being Element of M
	st Class(R,v) = Class(R,v9) & Class(R,w) = Class(R,w9)
	holds Class(R,vw) = Class(R,v9w9)
	proof
	let M be multMagma;
	let R be Equivalence_Relation of M;
	hereby
	assume A1: R is compatible;
	let v,v9,w,w9 be Element of M;
	assume A2: Class(R,v) = Class(R,v9) & Class(R,w) = Class(R,w9);
	per cases;
	suppose A3: M is empty;
	hence Class(R,vw) = {} .= Class(R,v9w9) by A3;
	end;
	suppose M is not empty; then
	v in Class(R,v9) & w in Class(R,w9) by A2,EQREL_1:23; then
	vw in Class(R,v9w9) by A1;
	hence Class(R,vw) = Class(R,v9w9) by EQREL_1:23;
	end;
	end;
	assume A4: for v,v9,w,w9 being Element of M
	st Class(R,v) = Class(R,v9) & Class(R,w) = Class(R,w9)
	holds Class(R,vw) = Class(R,v9w9);
	for v,v9,w,w9 being Element of M st v in Class(R,v9) & w in Class(R,w9)
	holds vw in Class(R,v9w9)
	proof
	let v,v9,w,w9 be Element of M;
	assume A5: v in Class(R,v9) & w in Class(R,w9);
	per cases;
	suppose M is empty; hence thesis by A5; end;
	suppose A6: M is not empty;
	Class(R,v9) = Class(R,v) &
	Class(R,w9) = Class(R,w) by A5,EQREL_1:23; then
	Class(R,vw) = Class(R,v9w9) by A4;
	hence vw in Class(R,v9w9) by A6,EQREL_1:23;
	end;
	end;
	hence R is compatible;
	end;

	definition
	let M be multMagma;
	let R be compatible Equivalence_Relation of M;
	func ClassOp R -> BinOp of Class R means :Def4:
	for x,y being Element of Class R, v,w being Element of M
	st x = Class(R,v) & y = Class(R,w) holds it.(x,y) = Class(R,v*w)
	if M is non empty otherwise it = {};
	correctness
	proof
	A1: M is not empty implies ex b being BinOp of Class R st
	for x, y being Element of Class R, v,w being Element of M st
	x = Class(R,v) & y = Class(R,w) holds b.(x,y) = Class(R,v*w)
	proof
	assume A2: M is not empty; then
	reconsider X = Class R as non empty set;
	defpred P[set,set,set] means for x,y being Element of Class R,
	v,w being Element of M st x=$1 & y=$2 & x = Class(R,v) & y = Class(R,w)
	holds $3 = Class(R,v*w);
	A3: for x,y being Element of X ex z being Element of X st P[x,y,z]
	proof
	let x,y be Element of X;
	x in Class R; then
	consider v be object such that
	A4: v in the carrier of M & x = Class(R,v) by EQREL_1:def 3;
	reconsider v as Element of M by A4;
	y in Class R; then
	consider w be object such that
	A5: w in the carrier of M & y = Class(R,w) by EQREL_1:def 3;
	reconsider w as Element of M by A5;
	reconsider z = Class(R,v*w) as Element of X by A2,EQREL_1:def 3;
	take z;
	thus thesis by A4,A5,Th2;
	end;
	consider b be Function of [:X,X:],X such that
	A6: for x,y being Element of X holds P[x,y,b.(x,y)]
	from BINOP_1:sch 3(A3);
	reconsider b as BinOp of Class R;
	take b;
	thus thesis by A6;
	end;
	A7: M is empty implies ex b being BinOp of Class R st b = {}
	proof
	assume M is empty; then
	the carrier of M is empty; then
	A8: Class R is empty;
	set b = the BinOp of Class R;
	take b;
	thus b = {} by A8;
	end;
	for b1, b2 being BinOp of Class R holds M is not empty &
	(for x,y being Element of Class R, v,w being Element of M
	st x = Class(R,v) & y = Class(R,w) holds b1.(x,y) = Class(R,v*w)) &
	(for x,y being Element of Class R, v,w being Element of M
	st x = Class(R,v) & y = Class(R,w) holds b2.(x,y) = Class(R,v*w))
	implies b1 = b2
	proof
	let b1,b2 be BinOp of Class R;
	assume M is not empty;
	assume A9: for x,y being Element of Class R, v,w being Element of M
	st x = Class(R,v) & y = Class(R,w) holds b1.(x,y) = Class(R,v*w);
	assume A10: for x,y being Element of Class R, v,w being Element of M
	st x = Class(R,v) & y = Class(R,w) holds b2.(x,y) = Class(R,v*w);
	for x,y being set st x in Class R & y in Class R
	holds b1.(x,y) = b2.(x,y)
	proof
	let x,y be set;
	assume A11: x in Class R; then
	reconsider x9=x as Element of Class R;
	assume A12: y in Class R; then
	reconsider y9=y as Element of Class R;
	consider v be object such that
	A13: v in the carrier of M & x9 = Class(R,v) by A11,EQREL_1:def 3;
	consider w be object such that
	A14:w in the carrier of M & y9 = Class(R,w) by A12,EQREL_1:def 3;
	reconsider v,w as Element of M by A13,A14;
	b1.(x9,y9) = Class(R,v*w) by A13,A14,A9;
	hence b1.(x,y) = b2.(x,y) by A13,A14,A10;
	end;
	hence thesis by BINOP_1:1;
	end;
	hence thesis by A1,A7;
	end;
	end;

	:: Ch I ?1.6 Def.11 Algebra I Bourbaki

	definition
	let M be multMagma;
	let R be compatible Equivalence_Relation of M;
	func M ./. R -> multMagma equals
	multMagma(# Class R, ClassOp R #);
	correctness;
	end;

	registration
	let M be multMagma;
	let R be compatible Equivalence_Relation of M;
	cluster M ./. R -> strict;
	correctness;
	end;

	registration
	let M be non empty multMagma;
	let R be compatible Equivalence_Relation of M;
	cluster M ./. R -> non empty;
	correctness;
	end;

	definition
	let M be non empty multMagma;
	let R be compatible Equivalence_Relation of M;
	func nat_hom R -> Function of M, M ./. R means :Def6:
	for v being Element of M holds it.v = Class(R,v);
	existence
	proof
	defpred P[object,object] means
	ex v being Element of M st $1=v & $2=Class(R,v);
	A1: for x being object st x in the carrier of M ex y being object st P[x,y]
	proof
	let x be object;
	assume x in the carrier of M;
	then reconsider v = x as Element of M;
	reconsider y = Class(R,v) as set;
	take y,v;
	thus thesis;
	end;
	consider f being Function such that
	A2: dom f = the carrier of M and
	A3: for x being object st x in the carrier of M
	holds P[x,f.x] from CLASSES1:sch 1(A1);
	rng f c= the carrier of M ./. R
	proof
	let x be object;
	assume x in rng f;
	then consider y be object such that
	A4: y in dom f and
	A5: f.y = x by FUNCT_1:def 3;
	consider v be Element of M such that
	A6: y = v & f.y = Class(R,v) by A2,A3,A4;
	thus thesis by A5,A6,EQREL_1:def 3;
	end;
	then reconsider f as Function of M, M ./. R
	by A2,FUNCT_2:def 1,RELSET_1:4;
	take f;
	let v be Element of M;
	ex w being Element of M st v = w & f.v = Class(R,w) by A3;
	hence thesis;
	end;
	uniqueness
	proof
	let f1,f2 be Function of M, M ./. R such that
	A7: for v being Element of M holds f1.v = Class(R,v) and
	A8: for v being Element of M holds f2.v = Class(R,v);
	now
	let v being Element of M;
	f1.v = Class(R,v) & f2.v = Class(R,v) by A7,A8;
	hence f1.v = f2.v;
	end;
	hence thesis by FUNCT_2:63;
	end;
	end;

	registration
	let M be non empty multMagma;
	let R be compatible Equivalence_Relation of M;
	cluster nat_hom R -> multiplicative;
	correctness
	proof
	for v,w being Element of M
	holds (nat_hom R).(vw) = (nat_hom R).v (nat_hom R).w
	proof
	let v,w be Element of M;
	(nat_hom R).v = Class(R,v) & (nat_hom R).w = Class(R,w) by Def6; then
	(ClassOp R).((nat_hom R).v,(nat_hom R).w) = Class(R,v*w) by Def4;
	hence (nat_hom R).(vw) = (nat_hom R).v (nat_hom R).w by Def6;
	end;
	hence thesis by GROUP_6:def 6;
	end;
	end;

	registration
	let M be non empty multMagma;
	let R be compatible Equivalence_Relation of M;
	cluster nat_hom R -> onto;
	correctness
	proof
	for y being object st y in the carrier of (M ./. R)
	holds y in rng nat_hom R
	proof
	let y be object;
	assume y in the carrier of (M ./. R); then
	consider x be object such that
	A1: x in the carrier of M & y = Class(R,x) by EQREL_1:def 3;
	reconsider x as Element of M by A1;
	the carrier of M = dom(nat_hom R) by FUNCT_2:def 1; then
	x in dom(nat_hom R) & (nat_hom R).x = Class(R,x) by Def6;
	hence y in rng nat_hom R by A1,FUNCT_1:def 3;
	end; then
	the carrier of (M ./. R) c= rng nat_hom R; then
	rng nat_hom R = the carrier of (M ./. R) by XBOOLE_0:def 10;
	hence thesis by FUNCT_2:def 3;
	end;
	end;

	definition
	let M be multMagma;
	mode Relators of M is [:the carrier of M,the carrier of M:]-valued Function;
	end;

	:: Ch I ?1.6 Algebra I Bourbaki

	definition
	let M be multMagma;
	let r be Relators of M;
	func equ_rel r -> Equivalence_Relation of M equals
	meet {R where R is compatible Equivalence_Relation of M:
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R,w)};
	correctness
	proof
	set X = {R where R is compatible Equivalence_Relation of M :
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R,w)};
	per cases;
	suppose M is empty; then
	A1: the carrier of M = {};
	for x being object st x in X holds x in {{}}
	proof
	let x be object;
	assume x in X; then
	consider R be compatible Equivalence_Relation of M such that
	A2: x=R &
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R,w);
	x is Subset of {} by A2,A1,ZFMISC_1:90;
	hence x in {{}} by TARSKI:def 1;
	end; then
	X c= {{}}; then
	X = {} or X = {{}} by ZFMISC_1:33;
	hence thesis by A1,OSALG_4:9,SETFAM_1:10,def 1;
	end;
	suppose A3: M is not empty;
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(nabla the carrier of M,w)
	proof
	let i be set;
	let v,w be Element of M;
	assume i in dom r & r.i = [v,w];
	Class(nabla the carrier of M,w) = the carrier of M by A3,EQREL_1:26;
	hence v in Class(nabla the carrier of M,w) by A3,SUBSET_1:def 1;
	end; then
	A4: nabla the carrier of M in X;
	for x being object st x in X
	holds x in bool [:the carrier of M,the carrier of M:]
	proof
	let x be object;
	assume x in X; then
	consider R be compatible Equivalence_Relation of M such that
	A5: x=R &
	for i being set, x,y being Element of M st i in dom r & r.i = [x,y]
	holds x in Class(R,y);
	thus thesis by A5;
	end; then
	reconsider X as Subset-Family of [:the carrier of M,the carrier of M:]
	by TARSKI:def 3;
	for Y being set st Y in X holds Y is Equivalence_Relation of M
	proof
	let Y be set;
	assume Y in X; then
	consider R be compatible Equivalence_Relation of M such that
	A6: Y=R &
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R,w);
	thus Y is Equivalence_Relation of M by A6;
	end;
	hence thesis by A4,EQREL_1:11;
	end;
	end;
	end;

	theorem Th3:
	for M being multMagma, r being Relators of M,
	R being compatible Equivalence_Relation of M
	st (for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R,w))
	holds equ_rel r c= R
	proof
	let M be multMagma;
	let r be Relators of M;
	let R be compatible Equivalence_Relation of M;
	assume A1: for i being set, v,w being Element of M
	st i in dom r & r.i = [v,w] holds v in Class(R,w);
	let X be object;
	R in {R9 where R9 is compatible Equivalence_Relation of M:
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R9,w)} by A1;
	hence thesis by SETFAM_1:def 1;
	end;

	registration
	let M be multMagma;
	let r be Relators of M;
	cluster equ_rel r -> compatible;
	coherence
	proof
	set X = {R where R is compatible Equivalence_Relation of M :
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R,w)};
	for v,v9,w,w9 being Element of M
	st v in Class(equ_rel r,v9) & w in Class(equ_rel r,w9)
	holds vw in Class(equ_rel r,v9w9)
	proof
	let v,v9,w,w9 be Element of M;
	assume v in Class(equ_rel r,v9); then
	A1: [v9,v] in equ_rel r by EQREL_1:18;
	assume w in Class(equ_rel r,w9); then
	A2: [w9,w] in equ_rel r by EQREL_1:18;
	per cases;
	suppose X = {}; hence thesis by A1,SETFAM_1:def 1; end;
	suppose A3: X <> {};
	for Y being set st Y in X holds [v9w9,vw] in Y
	proof
	let Y be set;
	assume A4: Y in X; then
	consider R be compatible Equivalence_Relation of M such that
	A5: Y = R &
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R,w);
	[v9,v] in Y by A1,A4,SETFAM_1:def 1; then
	A6: v in Class(R,v9) by A5,EQREL_1:18;
	[w9,w] in Y by A2,A4,SETFAM_1:def 1; then
	w in Class(R,w9) by A5,EQREL_1:18;
	then vw in Class(R,v9w9) by A6,Def3;
	hence [v9w9,vw] in Y by A5,EQREL_1:18;
	end; then
	[v9w9,vw] in meet X by A3,SETFAM_1:def 1;
	hence vw in Class(equ_rel r,v9w9) by EQREL_1:18;
	end;
	end;
	hence thesis;
	end;
	end;

	definition
	let X,Y be set;
	let f be Function of X,Y;
	func equ_kernel f -> Equivalence_Relation of X means :Def8:
	for x,y being object holds [x,y] in it iff x in X & y in X & f.x = f.y;
	existence
	proof
	defpred P[object,object] means f.$1 = f.$2;
	A1: for x being object st x in X holds P[x,x];
	A2: for x,y being object st P[x,y] holds P[y,x];
	A3: for x,y,z being object st P[x,y] & P[y,z] holds P[x,z];
	ex EqR being Equivalence_Relation of X st
	for x,y being object holds [x,y] in EqR iff x in X & y in X & P[x,y]
	from EQREL_1:sch 1(A1,A2,A3);
	hence thesis;
	end;
	uniqueness
	proof
	let R1, R2 be Equivalence_Relation of X;
	defpred P[object,object] means $1 in X & $2 in X & f.$1 = f.$2;
	assume for x,y being object holds [x,y] in R1
	iff x in X & y in X & f.x = f.y; then
	A4: for x,y being object holds [x,y] in R1 iff P[x,y];
	assume for x,y being object holds [x,y] in R2
	iff x in X & y in X & f.x = f.y; then
	A5: for x,y being object holds [x,y] in R2 iff P[x,y];
	thus R1 = R2 from RELAT_1:sch 2(A4,A5);
	end;
	end;

	reserve M,N for non empty multMagma,
	f for Function of M, N;

	theorem Th4:
	f is multiplicative implies equ_kernel f is compatible
	proof
	assume A1: f is multiplicative;
	set R = equ_kernel f;
	for v,v9,w,w9 being Element of M st v in Class(R,v9) & w in Class(R,w9) holds
	vw in Class(R,v9w9)
	proof
	let v,v9,w,w9 be Element of M;
	assume v in Class(R,v9); then
	A2: [v9,v] in R by EQREL_1:18;
	assume w in Class(R,w9); then
	[w9,w] in R by EQREL_1:18; then
	A3: f.w9 = f.w by Def8;
	f.(v9w9) = f.v9 f.w9 by A1,GROUP_6:def 6
	.= f.v * f.w by A2,A3,Def8
	.= f.(v*w) by A1,GROUP_6:def 6; then
	[v9w9,vw] in R by Def8;
	hence vw in Class(R,v9w9) by EQREL_1:18;
	end;
	hence equ_kernel f is compatible;
	end;

	theorem Th5:
	f is multiplicative implies
	ex r being Relators of M st equ_kernel f = equ_rel r
	proof
	assume A1: f is multiplicative;
	set I = {[v,w] where v,w is Element of M: f.v = f.w};
	set r = id I;
	for y being object st y in rng r
	holds y in [: the carrier of M, the carrier of M:]
	proof
	let y be object;
	assume y in rng r; then
	consider x be object such that
	A2: x in dom r & y = r.x by FUNCT_1:def 3;
	y = x by A2,FUNCT_1:17; then
	y in I by A2; then
	consider v9,w9 be Element of M such that
	A3: y = [v9,w9] & f.v9 = f.w9;
	thus thesis by A3,ZFMISC_1:def 2;
	end; then
	rng r c= [: the carrier of M, the carrier of M:]; then
	reconsider r as Relators of M by FUNCT_2:2;
	take r;
	reconsider R=equ_kernel f as compatible Equivalence_Relation of M by A1,Th4;
	A4: for i being set, v,w being Element of M
	st i in dom r & r.i = [v,w] holds v in Class(R,w)
	proof
	let i be set;
	let v,w be Element of M;
	assume A5: i in dom r & r.i = [v,w]; then
	A6: r.i = i by FUNCT_1:17;
	consider v9,w9 be Element of M such that
	A7: i=[v9,w9] & f.v9 = f.w9 by A5;
	[v,w] in equ_kernel f by A7,Def8,A5,A6;
	hence v in Class(R,w) by EQREL_1:19;
	end; then
	A8: equ_rel r c= R by Th3;
	for z being object st z in R holds z in equ_rel r
	proof
	let z be object;
	assume A9: z in R;
	then consider x,y be object such that
	A10: x in the carrier of M & y in the carrier of M &
	z = [x,y] by ZFMISC_1:def 2;
	A11: f.x = f.y by A9,A10,Def8;
	reconsider x,y as Element of M by A10;
	set X9 = {R9 where R9 is compatible Equivalence_Relation of M:
	for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
	holds v in Class(R9,w)};
	A12: R in X9 by A4;
	for Y being set st Y in X9 holds z in Y
	proof
	let Y be set;
	assume Y in X9; then
	consider R9 be compatible Equivalence_Relation of M such that
	A13: R9=Y & for i being set, v,w being Element of M
	st i in dom r & r.i = [v,w] holds v in Class(R9,w);
	set i = [x,y];
	A14: i in I by A11; then
	r.i = [x,y] by FUNCT_1:17; then
	x in Class(R9,y) by A14,A13;
	hence z in Y by A10,A13,EQREL_1:19;
	end;
	hence thesis by A12,SETFAM_1:def 1;
	end; then
	R c= equ_rel r;
	hence thesis by A8,XBOOLE_0:def 10;
	end;

	begin :: Submagmas and Stable Subsets

	definition
	let M be multMagma;
	mode multSubmagma of M -> multMagma means :Def9:
	the carrier of it c= the carrier of M &
	the multF of it = (the multF of M)\|\|the carrier of it;
	existence
	proof
	set S = the empty multMagma;
	reconsider A = the carrier of S as set;
	reconsider X = [: A, A :] as set;
	the multF of S = (the multF of M) \| {}
	.= (the multF of M) \| X by ZFMISC_1:90
	.= (the multF of M)\|\|the carrier of S by REALSET1:def 2;
	hence thesis by XBOOLE_1:2;
	end;
	end;

	registration
	let M be multMagma;
	cluster strict for multSubmagma of M;
	existence
	proof
	set N = multMagma(# the carrier of M, the multF of M #);
	the multF of N
	= (the multF of N)\|[:the carrier of N,the carrier of N:]
	.= (the multF of M)\|\|the carrier of N by REALSET1:def 2; then
	reconsider N as multSubmagma of M by Def9;
	take N;
	thus thesis;
	end;
	end;

	registration
	let M be non empty multMagma;
	cluster non empty for multSubmagma of M;
	existence
	proof
	set N = multMagma(# the carrier of M, the multF of M #);
	the multF of N
	= (the multF of N)\|[:the carrier of N,the carrier of N:]
	.= (the multF of M)\|\|the carrier of N by REALSET1:def 2; then
	reconsider N as multSubmagma of M by Def9;
	take N;
	thus thesis;
	end;
	end;

	reserve M for multMagma;
	reserve N,K for multSubmagma of M;

	:: like GROUP_2:64

	theorem Th6:
	N is multSubmagma of K & K is multSubmagma of N
	implies the multMagma of N = the multMagma of K
	proof
	assume that
	A1: N is multSubmagma of K and
	A2: K is multSubmagma of N;
	set A = the carrier of N;
	set B = the carrier of K;
	set f = the multF of N;
	set g = the multF of K;
	A3: A c= B & B c= A by A1,A2,Def9;
	f = g\|\|A by A1,Def9
	.= (f\|\|B)\|\|A by A2,Def9
	.= (f\|[:B,B:])\|\|A by REALSET1:def 2
	.= (f\|[:B,B:])\|[:A,A:] by REALSET1:def 2
	.= f\|[:B,B:]
	.= f\|\|B by REALSET1:def 2
	.= g by A2,Def9;
	hence thesis by A3,XBOOLE_0:def 10;
	end;

	theorem Th7:
	the carrier of N = the carrier of M
	implies the multMagma of N = the multMagma of M
	proof
	assume A1: the carrier of N = the carrier of M;
	per cases;
	suppose the carrier of M = {};
	hence thesis by A1;
	end;
	suppose the carrier of M <> {};
	A2: the multF of M
	= (the multF of M)\|[:the carrier of M,the carrier of M:]
	.= (the multF of M)\|\|(the carrier of M) by REALSET1:def 2; then
	reconsider M9=M as multSubmagma of M by Def9;
	the multF of M9 = (the multF of N)\|\|the carrier of M9 by A1,A2,Def9; then
	M9 is multSubmagma of N by A1,Def9;
	hence thesis by Th6;
	end;
	end;

	:: Ch I ?1.4 Def.6 Algebra I Bourbaki

	definition
	let M be multMagma;
	let A be Subset of M;
	attr A is stable means :Def10:
	for v,w being Element of M st v in A & w in A holds v*w in A;
	end;

	registration
	let M be multMagma;
	cluster stable for Subset of M;
	correctness
	proof
	reconsider A = the carrier of M as Subset of M by ZFMISC_1:def 1;
	take A;
	thus thesis;
	end;
	end;

	theorem Th8:
	the carrier of N is stable Subset of M
	proof
	for v,w being Element of M st v in the carrier of N & w in the carrier of N
	holds v*w in the carrier of N
	proof
	let v,w be Element of M;
	assume A1: v in the carrier of N & w in the carrier of N; then
	reconsider v9=v, w9=w as Element of N;
	A2: [v,w] in [:the carrier of N,the carrier of N:] by A1,ZFMISC_1:def 2;
	v9*w9 = (the multF of N).[v9,w9] by BINOP_1:def 1
	.= ((the multF of M)\|\|the carrier of N).[v9,w9] by Def9
	.= ((the multF of M)\|[:the carrier of N,the carrier of N:]).[v,w]
	by REALSET1:def 2
	.= (the multF of M).[v,w] by A2,FUNCT_1:49
	.= v*w by BINOP_1:def 1;
	hence v*w in the carrier of N by A1;
	end;
	hence the carrier of N is stable Subset of M by Def9,Def10;
	end;

	registration
	let M be multMagma;
	let N be multSubmagma of M;
	cluster the carrier of N -> stable for Subset of M;
	correctness by Th8;
	end;

	theorem Th9:
	for F being Function st
	(for i being set st i in dom F holds F.i is stable Subset of M) holds
	meet F is stable Subset of M
	proof
	let F be Function;
	assume A1: for i being set st i in dom F holds F.i is stable Subset of M;
	A2: for x being object st x in meet F holds x in the carrier of M
	proof
	let x be object;
	assume x in meet F; then
	A3: x in meet rng F by FUNCT_6:def 4;
	per cases;
	suppose dom F is empty; then
	F = {};
	hence thesis by A3,SETFAM_1:1;
	end;
	suppose dom F is not empty; then
	consider i be object such that
	A4: i in dom F by XBOOLE_0:def 1;
	meet rng F c= F.i by A4,FUNCT_1:3,SETFAM_1:3; then
	A5: x in F.i by A3;
	F.i is stable Subset of M by A1,A4;
	hence x in the carrier of M by A5;
	end;
	end;
	for v,w being Element of M st v in meet F & w in meet F holds v*w in meet F
	proof
	let v,w be Element of M;
	assume A6: v in meet F & w in meet F;
	per cases;
	suppose F = {}; then
	meet rng F = {} by SETFAM_1:1;
	hence thesis by A6,FUNCT_6:def 4;
	end;
	suppose A7: F <> {};
	A8: v in meet rng F & w in meet rng F by A6,FUNCT_6:def 4;
	for Y being set holds Y in rng F implies v*w in Y
	proof
	let Y be set;
	assume A9: Y in rng F; then
	A10: v in Y & w in Y by A8,SETFAM_1:def 1;
	consider i be object such that
	A11: i in dom F & Y = F.i by A9,FUNCT_1:def 3;
	Y is stable Subset of M by A1,A11;
	hence v*w in Y by A10,Def10;
	end; then
	v*w in meet rng F by A7,SETFAM_1:def 1;
	hence v*w in meet F by FUNCT_6:def 4;
	end;
	end;
	hence thesis by A2,Def10,TARSKI:def 3;
	end;

	reserve M,N for non empty multMagma,
	A for Subset of M,
	f,g for Function of M,N,
	X for stable Subset of M,
	Y for stable Subset of N;

	theorem
	A is stable iff A * A c= A
	proof
	hereby
	assume A1: A is stable;
	for x being object st x in A * A holds x in A
	proof
	let x be object;
	assume x in A * A; then
	consider v,w be Element of M such that
	A2: x = v * w & v in A & w in A by GROUP_2:8;
	thus x in A by A1,A2;
	end;
	hence A * A c= A;
	end;
	assume A3: A * A c= A;
	for v,w being Element of M st v in A & w in A holds v*w in A
	proof
	let v,w be Element of M;
	assume v in A & w in A; then
	vw in A A by GROUP_2:8;
	hence v*w in A by A3;
	end;
	hence A is stable;
	end;

	:: Ch I ?1.4 Pro.1 Algebra I Bourbaki

	theorem Th11:
	f is multiplicative implies f.:X is stable Subset of N
	proof
	assume A1: f is multiplicative;
	for v,w being Element of N st v in f.:X & w in f.:X holds v*w in f.:X
	proof
	let v,w be Element of N;
	assume v in f.:X; then
	consider v9 be object such that
	A2: v9 in dom f & v9 in X & v = f.v9 by FUNCT_1:def 6;
	assume w in f.:X; then
	consider w9 be object such that
	A3: w9 in dom f & w9 in X & w = f.w9 by FUNCT_1:def 6;
	reconsider v9,w9 as Element of M by A2,A3;
	v9*w9 in the carrier of M; then
	A4: v9*w9 in dom f by FUNCT_2:def 1;
	v9*w9 in X by A2,A3,Def10; then
	f.(v9*w9) in f.:X by A4,FUNCT_1:def 6;
	hence v*w in f.:X by A2,A3,A1,GROUP_6:def 6;
	end;
	hence f .: X is stable Subset of N by Def10;
	end;

	:: Ch I ?1.4 Pro.1 Algebra I Bourbaki

	theorem Th12:
	f is multiplicative implies f"Y is stable Subset of M
	proof
	assume A1: f is multiplicative;
	for v,w being Element of M st v in f"Y & w in f"Y holds v*w in f"Y
	proof
	let v,w be Element of M;
	assume v in f"Y; then
	A2: v in dom f & f.v in Y by FUNCT_1:def 7;
	assume w in f"Y; then
	A3: w in dom f & f.w in Y by FUNCT_1:def 7;
	v*w in the carrier of M; then
	A4: v*w in dom f by FUNCT_2:def 1;
	f.v*f.w in Y by A2,A3,Def10; then
	f.(v*w) in Y by A1,GROUP_6:def 6;
	hence v*w in f"Y by A4,FUNCT_1:def 7;
	end;
	hence f"Y is stable Subset of M by Def10;
	end;

	:: Ch I ?1.4 Pro.1 Algebra I Bourbaki

	theorem
	f is multiplicative & g is multiplicative
	implies {v where v is Element of M: f.v=g.v} is stable Subset of M
	proof
	assume A1: f is multiplicative;
	assume A2: g is multiplicative;
	set X = {v where v is Element of M: f.v=g.v};
	for x being object st x in X holds x in the carrier of M
	proof
	let x be object;
	assume x in X; then
	consider v be Element of M such that
	A3: x=v & f.v=g.v;
	thus x in the carrier of M by A3;
	end; then
	reconsider X as Subset of M by TARSKI:def 3;
	for v,w being Element of M st v in X & w in X holds v*w in X
	proof
	let v,w be Element of M;
	assume v in X; then
	consider v9 be Element of M such that
	A4: v=v9 & f.v9=g.v9;
	assume w in X; then
	consider w9 be Element of M such that
	A5: w=w9 & f.w9=g.w9;
	f.(vw) = g.vg.w by A4,A5,A1,GROUP_6:def 6
	.= g.(v*w) by A2,GROUP_6:def 6;
	hence v*w in X;
	end;
	hence thesis by Def10;
	end;

	:: Ch I ?1.4 Def.6 Algebra I Bourbaki

	definition
	let M be multMagma;
	let A be stable Subset of M;
	func the_mult_induced_by A -> BinOp of A equals
	(the multF of M) \|\| A;
	correctness
	proof
	for x being set holds x in [:A,A:] implies (the multF of M).x in A
	proof
	let x be set;
	assume x in [:A,A:]; then
	consider v,w be object such that
	A1: v in A & w in A & x = [v,w] by ZFMISC_1:def 2;
	reconsider v,w as Element of M by A1;
	v*w in A by A1,Def10;
	hence (the multF of M).x in A by A1,BINOP_1:def 1;
	end; then
	A is Preserv of (the multF of M) by REALSET1:def 1;
	hence thesis by REALSET1:2;
	end;
	end;

	:: like GROUP_4:def 5

	definition
	let M be multMagma;
	let A be Subset of M;
	func the_submagma_generated_by A -> strict multSubmagma of M means :Def12:
	A c= the carrier of it & for N being strict multSubmagma of M
	st A c= the carrier of N holds it is multSubmagma of N;
	existence
	proof
	defpred P[set] means ex H being strict multSubmagma of M st
	$1 = the carrier of H & A c= $1;
	consider X be set such that
	A1: for Y being set holds Y in X iff Y in bool the carrier of M & P[Y]
	from XFAMILY:sch 1;
	set F = id X;
	set A1 = meet id X;
	for x being set st x in dom F holds F.x is stable Subset of M
	proof
	let x be set;
	assume A2: x in dom F; then
	x in bool the carrier of M & P[x] by A1;
	hence thesis by A2,FUNCT_1:18;
	end; then
	reconsider A1 as stable Subset of M by Th9;
	set N1 = multMagma(# A1, the_mult_induced_by A1 #);
	take N1;
	per cases;
	suppose A3: X = {};
	A4: the carrier of M in bool the carrier of M by ZFMISC_1:def 1;
	ex H being strict multSubmagma of M st
	the carrier of M= the carrier of H & A c= the carrier of M
	proof
	the multF of M = (the multF of M)\|[:the carrier of M,the carrier of M:]
	; then
	the multF of M = (the multF of M)\|\|the carrier of M
	by REALSET1:def 2; then
	reconsider H = the multMagma of M as strict multSubmagma of M by Def9;
	take H;
	thus the carrier of M = the carrier of H;
	thus A c= the carrier of M;
	end;
	hence thesis by A3,A4,A1;
	end;
	suppose A5: X <> {};
	A6: for x being object st x in A holds x in A1
	proof
	let x be object;
	assume A7: x in A;
	for Y being set st Y in X holds x in Y
	proof
	let Y be set;
	assume Y in X; then
	consider H be strict multSubmagma of M such that
	A8: Y = the carrier of H & A c= Y by A1;
	thus x in Y by A8,A7;
	end; then
	x in meet X by A5,SETFAM_1:def 1; then
	x in meet rng id X;
	hence thesis by FUNCT_6:def 4;
	end;
	for N being strict multSubmagma of M
	st A c= the carrier of N holds N1 is multSubmagma of N
	proof
	let N be strict multSubmagma of M;
	assume A9: A c= the carrier of N;
	for x being object
	st x in the carrier of N1 holds x in the carrier of N
	proof
	let x be object;
	assume x in the carrier of N1; then
	x in meet rng id X by FUNCT_6:def 4; then
	A10: x in meet X;
	the carrier of N c= the carrier of M by Def9;
	then the carrier of N in X by A1,A9;
	hence x in the carrier of N by A10,SETFAM_1:def 1;
	end; then
	A11: the carrier of N1 c= the carrier of N;
	A12: (the multF of M)\|[:the carrier of N,the carrier of N:]
	= (the multF of M)\|\|the carrier of N by REALSET1:def 2
	.= the multF of N by Def9;
	the multF of N1
	= (the multF of M)\|[:the carrier of N1,the carrier of N1:]
	by REALSET1:def 2
	.= ((the multF of M)\|[:the carrier of N,the carrier of N:])
	\|[:the carrier of N1,the carrier of N1:]
	by A11,RELAT_1:74,ZFMISC_1:96
	.= (the multF of N)\|\|the carrier of N1 by A12,REALSET1:def 2;
	hence N1 is multSubmagma of N by A11,Def9;
	end;
	hence thesis by A6,Def9;
	end;
	end;
	uniqueness
	proof
	let H1,H2 be strict multSubmagma of M;
	assume A c= the carrier of H1 & (for N being strict multSubmagma of M
	st A c= the carrier of N holds H1 is multSubmagma of N) &
	A c= the carrier of H2 & (for N being strict multSubmagma of M
	st A c= the carrier of N holds H2 is multSubmagma of N);
	then H1 is multSubmagma of H2 & H2 is multSubmagma of H1;
	hence thesis by Th6;
	end;
	end;

	theorem Th14:
	for M being multMagma, A being Subset of M
	holds A is empty iff the_submagma_generated_by A is empty
	proof
	let M be multMagma;
	let A be Subset of M;
	hereby
	assume A1: A is empty;
	then
	for v,w being Element of M st v in A & w in A holds v*w in A; then
	reconsider A9=A as stable Subset of M by Def10;
	reconsider N=multMagma(# A9, the_mult_induced_by A9 #)
	as strict multSubmagma of M by Def9;
	the_submagma_generated_by A is multSubmagma of N by Def12; then
	the carrier of the_submagma_generated_by A c= the carrier of N by Def9;
	hence the_submagma_generated_by A is empty by A1;
	end;
	assume the_submagma_generated_by A is empty; then
	the carrier of the_submagma_generated_by A = {}; then
	A c= {} by Def12;
	hence A is empty;
	end;

	registration
	let M be multMagma;
	let A be empty Subset of M;
	cluster the_submagma_generated_by A -> empty;
	correctness by Th14;
	end;

	:: Ch I ?1.4 Pro.1 Algebra I Bourbaki

	theorem Th15:
	for M,N being non empty multMagma, f being Function of M,N,
	X being Subset of M st f is multiplicative holds
	f.:the carrier of the_submagma_generated_by X =
	the carrier of the_submagma_generated_by f.:X
	proof
	let M,N be non empty multMagma;
	let f be Function of M,N;
	let X be Subset of M;
	assume A1: f is multiplicative;
	set X9 = the_submagma_generated_by X;
	set A = f.:the carrier of X9;
	the carrier of X9 is stable Subset of M by Th8; then
	reconsider A as stable Subset of N by A1,Th11;
	set Y9 = the_submagma_generated_by f.:X;
	set B = f"the carrier of Y9;
	the carrier of Y9 is stable Subset of N by Th8; then
	reconsider B as stable Subset of M by A1,Th12;
	A2: f.:X c= the carrier of Y9 & for N1 being strict multSubmagma of N
	st f.:X c= the carrier of N1
	holds Y9 is multSubmagma of N1 by Def12;
	reconsider N1 = multMagma(# A, the_mult_induced_by A #) as
	strict multSubmagma of N by Def9;
	X c= the carrier of X9 by Def12; then
	Y9 is multSubmagma of N1 by A2,RELAT_1:123; then
	A3: the carrier of Y9 c= A by Def9;
	A4: X c= the carrier of X9 & for M1 being strict multSubmagma of M
	st X c= the carrier of M1
	holds X9 is multSubmagma of M1 by Def12;
	reconsider M1 = multMagma(# B, the_mult_induced_by B #) as
	strict multSubmagma of M by Def9;
	A5: f.:(f"the carrier of Y9) c= the carrier of Y9 by FUNCT_1:75;
	f.:X c= the carrier of the_submagma_generated_by f.:X by Def12; then
	A6: f"(f.:X) c= f"the carrier of the_submagma_generated_by f.:X
	by RELAT_1:143;
	X c= the carrier of M; then
	X c= dom f by FUNCT_2:def 1; then
	X c= f"(f.:X) by FUNCT_1:76; then
	X9 is multSubmagma of M1 by A4,A6,XBOOLE_1:1; then
	the carrier of X9 c= B by Def9; then
	A c= f.:(f"the carrier of Y9) by RELAT_1:123; then
	A c= the carrier of Y9 by A5;
	hence thesis by A3,XBOOLE_0:def 10;
	end;

	begin :: Free Magmas
	:: Ch I ?7.1 Algebra I Bourbaki

	definition
	let X be set;
	defpred P[object,object] means
	for fs being XFinSequence of bool the_universe_of(X \/ NAT)
	st $1=fs holds
	(dom fs = 0 implies $2 = {}) &
	(dom fs = 1 implies $2 = X) &
	for n being Nat st n>=2 & dom fs = n holds
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	fs1.p = [: fs.p, fs.(n-p) :] ) & $2 = Union disjoin fs1;
	A1: for e being object st e in (bool the_universe_of(X \/ NAT))^omega
	ex u being object st P[e,u]
	proof
	let e be object;
	assume e in (bool the_universe_of(X \/ NAT))^omega; then
	reconsider fs = e as
	XFinSequence of bool the_universe_of(X \/ NAT) by AFINSQ_1:def 7;
	dom fs = 0 or dom fs + 1 > 0+1 by XREAL_1:6; then
	dom fs = 0 or dom fs >= 1 by NAT_1:13; then
	dom fs = 0 or dom fs = 1 or dom fs > 1 by XXREAL_0:1; then
	A2: dom fs = 0 or dom fs = 1 or dom fs + 1 > 1+1 by XREAL_1:6;
	per cases by A2,NAT_1:13;
	suppose A3: dom fs = 0; set u = {};
	take u; thus P[e,u] by A3; end;
	suppose A4: dom fs = 1; set u = X;
	take u; thus P[e,u] by A4; end;
	suppose A5: dom fs >= 2;
	reconsider n = dom fs as Nat;
	reconsider n9= n -' 1 as Nat;
	n-1 >= 2-1 by A5,XREAL_1:9; then
	A6: n9 = n-1 by XREAL_0:def 2;
	defpred P2[set,object] means for p being Nat
	st p>=1 & p<=n-1 & $1=p holds $2 = [: fs.p, fs.(n-p) :];
	A7: for k being Nat st k in Seg n9 ex x being object st P2[k,x]
	proof
	let k be Nat;
	assume k in Seg n9;
	set x = [: fs.k, fs.(n-k) :];
	take x;
	thus thesis;
	end;
	consider fs1 be FinSequence such that
	A8: dom fs1 = Seg n9 & for k being Nat st k in Seg n9 holds P2[k,fs1.k]
	from FINSEQ_1:sch 1(A7);
	set u = Union disjoin fs1;
	take u;
	A9: len fs1 = n-1 by A6,A8,FINSEQ_1:def 3;
	for p being Nat st p>=1 & p<=n-1 holds
	fs1.p = [: fs.p, fs.(n-p) :] by A8,A6,FINSEQ_1:1;
	hence P[e,u] by A5,A9;
	end;
	end;
	consider F be Function such that
	A10: dom F = (bool the_universe_of(X \/ NAT))^omega &
	for e being object st e in (bool the_universe_of(X \/ NAT))^omega
	holds P[e,F.e] from CLASSES1:sch 1(A1);
	A11: for e being object st e in (bool the_universe_of(X \/ NAT))^omega
	holds F.e in bool the_universe_of(X \/ NAT)
	proof
	let e be object;
	assume A12: e in (bool the_universe_of(X \/ NAT))^omega;
	then reconsider fs=e as XFinSequence of bool the_universe_of(X \/ NAT)
	by AFINSQ_1:def 7;
	A13: (dom fs = 0 implies F.e = {}) & (dom fs = 1 implies F.e = X) &
	for n being Nat st n>=2 & dom fs = n holds
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	fs1.p = [: fs.p, fs.(n-p) :] ) & F.e = Union disjoin fs1 by A12,A10;
	dom fs = 0 or dom fs + 1 > 0+1 by XREAL_1:6; then
	dom fs = 0 or dom fs >= 1 by NAT_1:13; then
	dom fs = 0 or dom fs = 1 or dom fs > 1 by XXREAL_0:1; then
	A14: dom fs = 0 or dom fs = 1 or dom fs + 1 > 1+1 by XREAL_1:6;
	per cases by A14,NAT_1:13;
	suppose A15: dom fs = 0;
	{} c= the_universe_of(X \/ NAT);
	hence F.e in bool the_universe_of(X \/ NAT) by A15,A13;
	end;
	suppose dom fs = 1; then
	A16: F.e = X by A12,A10;
	for x being object st x in X holds
	x in Tarski-Class the_transitive-closure_of(X \/ NAT)
	proof
	let x be object;
	reconsider xx=x as set by TARSKI:1;
	assume x in X; then
	xx c= (union X) \/ union NAT by XBOOLE_1:10,ZFMISC_1:74; then
	A17: xx c= union(X \/ NAT) by ZFMISC_1:78;
	A18: the_transitive-closure_of(X \/ NAT) in
	Tarski-Class the_transitive-closure_of(X \/ NAT) by CLASSES1:2;
	A19: union(X \/ NAT) c= union the_transitive-closure_of(X \/ NAT)
	by CLASSES1:52,ZFMISC_1:77;
	union the_transitive-closure_of(X \/ NAT)
	c= the_transitive-closure_of(X \/ NAT)
	by CLASSES1:48,51; then
	union(X \/ NAT) c= the_transitive-closure_of(X \/ NAT)
	by A19;
	hence thesis by A18,A17,CLASSES1:3,XBOOLE_1:1;
	end; then
	X c= Tarski-Class the_transitive-closure_of(X \/ NAT); then
	X c= the_universe_of(X \/ NAT) by YELLOW_6:def 1;
	hence F.e in bool the_universe_of(X \/ NAT) by A16;
	end;
	suppose A20: dom fs >= 2;
	set n=dom fs;
	consider fs1 be FinSequence such that
	A21: len fs1 = n-1 & (for p being Nat st p>=1 & p<=n-1 holds
	fs1.p = [: fs.p, fs.(n-p) :]) &
	F.e = Union disjoin fs1 by A20,A12,A10;
	reconsider n9= n -' 1 as Nat;
	n-1 >= 2-1 by A20,XREAL_1:9; then
	A22: n9 = n-1 by XREAL_0:def 2;
	A23: for p being Nat st p>=1 & p<=n-1 holds
	fs1.p c= the_universe_of(X \/ NAT)
	proof
	let p be Nat;
	assume A24: p>=1 & p<=n-1; then
	A25: fs1.p = [: fs.p, fs.(n-p) :] by A21;
	A26: p in Seg n9 by A22,A24,FINSEQ_1:1;
	-p <= -1 & -p >= -(n-1) by A24,XREAL_1:24; then
	A27: -p +n <= -1+n & -p+n >= -(n-1)+n by XREAL_1:6; then
	A28: n-p <= n-'1 & n-p >= 1 by XREAL_0:def 2;
	A29: n-p = n-'p by A27,XREAL_0:def 2; then
	A30: n-'p in Seg n9 by A28,FINSEQ_1:1;
	A31: Seg n9 c= Segm(n9+1) by AFINSQ_1:3; then
	A32: fs.p in rng fs by A26,A22,FUNCT_1:3;
	fs.(n-p) in rng fs by A29,A31,A22,A30,FUNCT_1:3;
	hence fs1.p c= the_universe_of(X \/ NAT) by A25,A32,Th1;
	end;
	for x being set st x in rng disjoin fs1
	holds x c= the_universe_of(X \/ NAT)
	proof
	let x be set;
	assume x in rng disjoin fs1; then
	consider p be object such that
	A33: p in dom disjoin fs1 & x = (disjoin fs1).p by FUNCT_1:def 3;
	A34: p in dom fs1 by A33,CARD_3:def 3;
	then A35: x = [:fs1.p,{p}:] by A33,CARD_3:def 3;
	A36: p in Seg n9 by A21,A22,A34,FINSEQ_1:def 3;
	reconsider p as Nat by A34;
	p>=1 & p<=n-1 by A22,A36,FINSEQ_1:1; then
	A37: fs1.p c= the_universe_of(X \/ NAT) by A23;
	A38: for y being set st y in {p} holds y in NAT
	proof
	let y be set;
	assume y in {p}; then
	y = p by TARSKI:def 1;
	hence y in NAT by ORDINAL1:def 12;
	end;
	for x being object st x in {p} holds
	x in Tarski-Class the_transitive-closure_of(X \/ NAT)
	proof
	let x be object;
	reconsider xx=x as set by TARSKI:1;
	assume x in {p}; then
	x in NAT by A38; then
	xx c= (union X) \/ union NAT by XBOOLE_1:10,ZFMISC_1:74; then
	A39: xx c= union(X \/ NAT) by ZFMISC_1:78;
	A40: the_transitive-closure_of(X \/ NAT) in
	Tarski-Class the_transitive-closure_of(X \/ NAT) by CLASSES1:2;
	A41: union(X \/ NAT) c= union the_transitive-closure_of(X \/ NAT)
	by CLASSES1:52,ZFMISC_1:77;
	union the_transitive-closure_of(X \/ NAT)
	c= the_transitive-closure_of(X \/ NAT)
	by CLASSES1:48,51; then
	union(X \/ NAT) c= the_transitive-closure_of(X \/ NAT)
	by A41;
	hence thesis by A40,A39,CLASSES1:3,XBOOLE_1:1;
	end; then
	{p} c= Tarski-Class the_transitive-closure_of(X \/ NAT); then
	{p} c= the_universe_of(X \/ NAT) by YELLOW_6:def 1;
	hence thesis by A35,A37,Th1;
	end; then
	union (rng disjoin fs1) c= the_universe_of(X \/ NAT)
	by ZFMISC_1:76; then
	union (rng disjoin fs1) in bool the_universe_of(X \/ NAT);
	hence thesis by A21,CARD_3:def 4;
	end;
	end;
	func free_magma_seq X ->
	sequence of bool the_universe_of(X \/ NAT) means :Def13:
	it.0 = {} & it.1 = X & for n being Nat st n>=2 holds
	ex fs being FinSequence st len fs = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	fs.p = [: it.p, it.(n-p) :] ) & it.n = Union disjoin fs;
	existence
	proof
	reconsider F as Function of (bool the_universe_of(X \/ NAT))^omega,
	bool the_universe_of(X \/ NAT) by A11,A10,FUNCT_2:3;
	deffunc FX(XFinSequence of bool the_universe_of(X \/ NAT)) = F.$1;
	consider f be sequence of bool the_universe_of(X \/ NAT) such that
	A42: for n being Nat holds f.n = FX(f\|n)
	from FuncRecursiveExist2;
	take f;
	A43: {} in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:43;
	A44: dom {} = {};
	thus f.0 = F.(f\|{}) by A42 .= {} by A43,A44,A10;
	1 c= NAT; then
	1 c= dom f by FUNCT_2:def 1; then
	A45: dom(f\|1) = 1 by RELAT_1:62;
	A46: f\|1 in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:42;
	thus f.1 = F.(f\|1) by A42 .= X by A45,A46,A10;
	let n be Nat;
	assume A47: n >= 2;
	n c= NAT; then
	n c= dom f by FUNCT_2:def 1; then
	A48: dom(f\|n) = n by RELAT_1:62;
	f\|n in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:42;
	then consider fs1 be FinSequence such that
	A49: len fs1 = n-1 & (for p being Nat st p>=1 & p<=n-1 holds
	fs1.p = [: (f\|n).p, (f\|n).(n-p) :] ) &
	F.(f\|n) = Union disjoin fs1 by A47,A48,A10;
	take fs1;
	thus len fs1 = n-1 by A49;
	thus for p being Nat st p>=1 & p<=n-1
	holds fs1.p = [: f.p, f.(n - p):]
	proof
	let p be Nat;
	assume A50: p >= 1 & p <= n - 1;
	set n9 = n-'1;
	n-1 >= 2-1 by A47,XREAL_1:9; then
	A51: n9 = n-1 by XREAL_0:def 2; then
	A52: p in Seg n9 by A50,FINSEQ_1:1;
	Seg n9 c= Segm(n9+1) by AFINSQ_1:3; then
	A53: (f\|n).p = f.p by A51,A52,FUNCT_1:49;
	-p <= -1 & -p >= -(n-1) by A50,XREAL_1:24; then
	A54: -p +n <= -1+n & -p+n >= -(n-1)+n by XREAL_1:6; then
	A55: n-p <= n-'1 & n-p >= 1 by XREAL_0:def 2;
	A56: n-p = n-'p by A54,XREAL_0:def 2; then
	A57: n-'p in Seg n9 by A55,FINSEQ_1:1;
	A58: Seg n9 c= Segm(n9+1) by AFINSQ_1:3;
	thus fs1.p = [: (f\|n).p, (f\|n).(n-p) :] by A50,A49
	.= [: f.p, f.(n-p):] by A53,A58,A56,A51,A57,FUNCT_1:49;
	end;
	thus f.n = Union disjoin fs1 by A49,A42;
	end;
	uniqueness
	proof
	let f1, f2 be sequence of bool the_universe_of(X \/ NAT);
	assume A59: f1.0 = {};
	assume A60: f1.1 = X;
	assume A61: for n being Nat st n >= 2 holds
	ex fs being FinSequence st len fs = n - 1 &
	(for p being Nat st p >= 1 & p <= n - 1 holds
	fs.p = [: f1.p, f1.(n-p) :] ) & f1.n = Union disjoin fs;
	assume A62: f2.0 = {};
	assume A63: f2.1 = X;
	assume A64: for n being Nat st n >= 2 holds
	ex fs being FinSequence st len fs = n - 1 &
	(for p being Nat st p >= 1 & p <= n - 1 holds
	fs.p = [: f2.p, f2.(n-p):] ) & f2.n = Union disjoin fs;
	{} in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:43; then
	A65: P[{},F.{}] & {} is XFinSequence of bool the_universe_of(X \/ NAT)
	by A10,AFINSQ_1:42;
	A66: dom {} = {};
	reconsider F as Function of (bool the_universe_of(X \/ NAT))^omega,
	bool the_universe_of(X \/ NAT) by A11,A10,FUNCT_2:3;
	deffunc FX(XFinSequence of bool the_universe_of(X \/ NAT)) = F.$1;
	A67: for n being Nat holds f1.n = FX(f1\|n)
	proof
	let n be Nat;
	n = 0 or n + 1 > 0+1 by XREAL_1:6; then
	n = 0 or n >= 1 by NAT_1:13; then
	n = 0 or n = 1 or n > 1 by XXREAL_0:1; then
	A68: n = 0 or n = 1 or n + 1 > 1+1 by XREAL_1:6;
	per cases by A68,NAT_1:13;
	suppose A69: n=0;
	hence f1.n = F.{} by A65,A66,A59
	.= FX(f1\|n) by A69;
	end;
	suppose A70: n=1;
	1 c= NAT; then
	1 c= dom f1 by FUNCT_2:def 1; then
	A71: dom(f1\|1) = 1 by RELAT_1:62;
	f1\|1 in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:42;
	hence f1.n = FX(f1\|n) by A70,A71,A10,A60;
	end;
	suppose A72: n>=2;
	n c= NAT; then
	n c= dom f1 by FUNCT_2:def 1; then
	A73: dom(f1\|n) = n by RELAT_1:62;
	f1\|n in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:42; then
	consider fs1 be FinSequence such that
	A74: len fs1 = n-1 & (for p being Nat st p>=1 & p<=n-1 holds
	fs1.p = [: (f1\|n).p, (f1\|n).(n-p) :] ) &
	F.(f1\|n) = Union disjoin fs1 by A72,A73,A10;
	consider fs2 be FinSequence such that
	A75: len fs2 = n - 1 &
	(for p being Nat st p >= 1 & p <= n - 1 holds
	fs2.p = [: f1.p, f1.(n - p) :] ) &
	f1.n = Union disjoin fs2 by A72,A61;
	for p being Nat st 1 <= p & p <= len fs1 holds fs1.p = fs2.p
	proof
	let p be Nat;
	assume A76: 1 <= p & p <= len fs1;
	then A77: fs1.p = [: (f1\|n).p, (f1\|n).(n-p) :] by A74;
	A78: fs2.p = [: f1.p, f1.(n-p):] by A76,A74,A75;
	set n9 = n-'1;
	n-1 >= 2-1 by A72,XREAL_1:9; then
	A79: n9 = n-1 by XREAL_0:def 2; then
	A80: p in Seg n9 by A76,A74,FINSEQ_1:1;
	A81: Seg n9 c= Segm(n9+1) by AFINSQ_1:3;
	-p <= -1 & -p >= -(n-1) by A76,A74,XREAL_1:24; then
	A82: -p +n <= -1+n & -p+n >= -(n-1)+n by XREAL_1:6; then
	A83: n-p <= n-'1 & n-p >= 1 by XREAL_0:def 2;
	A84: n-p = n-'p by A82,XREAL_0:def 2; then
	A85: n-'p in Seg n9 by A83,FINSEQ_1:1;
	Seg n9 c= Segm(n9+1) by AFINSQ_1:3; then
	(f1\|n).(n-p) = f1.(n-p) by A84,A79,A85,FUNCT_1:49;
	hence fs1.p = fs2.p by A81,A77,A78,A79,A80,FUNCT_1:49;
	end;
	hence f1.n = FX(f1\|n) by A74,A75,FINSEQ_1:14;
	end;
	end;
	A86: for n being Nat holds f2.n = FX(f2\|n)
	proof
	let n be Nat;
	n = 0 or n + 1 > 0+1 by XREAL_1:6; then
	n = 0 or n >= 1 by NAT_1:13; then
	n = 0 or n = 1 or n > 1 by XXREAL_0:1; then
	A87: n = 0 or n = 1 or n + 1 > 1+1 by XREAL_1:6;
	per cases by A87,NAT_1:13;
	suppose A88: n=0; hence f2.n = F.{} by A65,A66,A62 .= FX(f2\|n) by A88;
	end;
	suppose A89: n=1;
	1 c= NAT; then
	1 c= dom f2 by FUNCT_2:def 1; then
	A90: dom(f2\|1) = 1 by RELAT_1:62;
	f2\|1 in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:42;
	hence f2.n = FX(f2\|n) by A90,A10,A89,A63;
	end;
	suppose A91: n>=2;
	n c= NAT; then
	n c= dom f2 by FUNCT_2:def 1; then
	A92: dom(f2\|n) = n by RELAT_1:62;
	f2\|n in (bool the_universe_of(X \/ NAT))^omega by AFINSQ_1:42; then
	consider fs1 be FinSequence such that
	A93: len fs1 = n-1 & (for p being Nat st p>=1 & p<=n-1 holds
	fs1.p = [: (f2\|n).p, (f2\|n).(n-p) :] ) &
	F.(f2\|n) = Union disjoin fs1 by A91,A92,A10;
	consider fs2 be FinSequence such that
	A94: len fs2 = n - 1 &
	(for p being Nat st p >= 1 & p <= n - 1 holds
	fs2.p = [: f2.p, f2.(n-p):] ) & f2.n = Union disjoin fs2 by A91,A64;
	for p being Nat st 1 <= p & p <= len fs1 holds fs1.p = fs2.p
	proof
	let p be Nat;
	assume A95: 1 <= p & p <= len fs1; then
	A96: fs1.p = [: (f2\|n).p, (f2\|n).(n-p) :] by A93;
	A97: fs2.p = [: f2.p, f2.(n-p):] by A95,A93,A94;
	set n9 = n-'1;
	n-1 >= 2-1 by A91,XREAL_1:9; then
	A98: n9 = n-1 by XREAL_0:def 2; then
	A99: p in Seg n9 by A95,A93,FINSEQ_1:1;
	A100: Seg n9 c= Segm(n9+1) by AFINSQ_1:3;
	-p <= -1 & -p >= -(n-1) by A95,A93,XREAL_1:24; then
	A101: -p +n <= -1+n & -p+n >= -(n-1)+n by XREAL_1:6; then
	A102: n-p <= n-'1 & n-p >= 1 by XREAL_0:def 2;
	A103: n-p = n-'p by A101,XREAL_0:def 2; then
	A104: n-'p in Seg n9 by A102,FINSEQ_1:1;
	Seg n9 c= Segm(n9+1) by AFINSQ_1:3; then
	(f2\|n).(n-p) = f2.(n-p) by A104,A103,A98,FUNCT_1:49;
	hence fs1.p = fs2.p by A100,A96,A97,A98,A99,FUNCT_1:49;
	end;
	hence f2.n = FX(f2\|n) by A94,A93,FINSEQ_1:14;
	end;
	end;
	f1=f2 from FuncRecursiveUniqu2(A67,A86);
	hence thesis;
	end;
	end;

	definition
	let X be set;
	let n be Nat;
	func free_magma(X,n) -> set equals (free_magma_seq X).n;
	correctness;
	end;

	registration
	let X be non empty set;
	let n be non zero Nat;
	cluster free_magma(X,n) -> non empty;
	correctness
	proof
	defpred P[Nat] means $1 = 0 or (free_magma_seq X).$1 <> {};
	A1: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
	proof
	let k be Nat;
	assume A2: for n being Nat st n < k holds P[n];
	k = 0 or k + 1 > 0+1 by XREAL_1:6; then
	k = 0 or k >= 1 by NAT_1:13; then
	k = 0 or k = 1 or k > 1 by XXREAL_0:1; then
	A3: k = 0 or k = 1 or k + 1 > 1+1 by XREAL_1:6;
	per cases by A3,NAT_1:13;
	suppose k=0; hence P[k]; end;
	suppose k=1; hence P[k] by Def13; end;
	suppose A4: k>=2; then
	consider fs be FinSequence such that
	A5: len fs = k-1 & (for p being Nat st p>=1 & p<=k-1 holds
	fs.p = [: (free_magma_seq X).p, (free_magma_seq X).(k-p) :] ) &
	(free_magma_seq X).k = Union disjoin fs by Def13;
	A6: 2-1<=k-1 by A4,XREAL_1:9; then
	1 in Seg len fs by A5,FINSEQ_1:1; then
	A7: 1 in dom fs by FINSEQ_1:def 3; then
	A8: 1 in dom disjoin fs by CARD_3:def 3;
	A9: (disjoin fs).1 = [:fs.1,{1}:] by A7,CARD_3:def 3;
	A10: fs.1=[:(free_magma_seq X).1,(free_magma_seq X).(k-1) :] by A5,A6;
	1+1 <= k-1+1 by A4; then
	1 < k by NAT_1:13; then
	A11: (free_magma_seq X).1 <> {} by A2;
	A12: -1+k < 0+k by XREAL_1:8;
	k-1 in NAT by A6,INT_1:3; then
	reconsider k9=k-1 as Nat;
	(free_magma_seq X).k9 <> {} by A12,A6,A2;
	then consider x be object such that
	A13: x in [:fs.1,{1}:] by A11,A10,XBOOLE_0:def 1;
	[:fs.1,{1}:] c= union rng disjoin fs by A9,A8,FUNCT_1:3,ZFMISC_1:74;
	hence P[k] by A13,A5,CARD_3:def 4;
	end;
	end;
	for n being Nat holds P[n] from NAT_1:sch 4(A1);
	hence thesis;
	end;
	end;

	reserve X for set;

	theorem
	free_magma(X,0) = {} by Def13;

	theorem
	free_magma(X,1) = X by Def13;

	theorem Th18:
	free_magma(X,2) = [:[:X,X:],{1}:]
	proof
	consider fs be FinSequence such that
	A1: len fs = 2-1 & (for p being Nat st p>=1 & p<=2-1 holds
	fs.p = [: (free_magma_seq X).p, (free_magma_seq X).(2-p) :] ) &
	(free_magma_seq X).2 = Union disjoin fs by Def13;
	A2: fs.1 = [: (free_magma_seq X).1, (free_magma_seq X).(2-1) :] by A1
	.= [: free_magma(X,1), X :] by Def13 .= [: X, X :] by Def13; then
	A3: fs = <* [:X,X:] *> by A1,FINSEQ_1:40;
	A4: for y being object
	holds y in union rng disjoin fs iff y in [:[:X,X:],{1}:]
	proof
	let y be object;
	hereby
	assume y in union rng disjoin fs; then
	consider Y be set such that
	A5: y in Y & Y in rng disjoin fs by TARSKI:def 4;
	consider x be object such that
	A6: x in dom disjoin fs & Y = (disjoin fs).x by A5,FUNCT_1:def 3;
	A7: x in dom fs by A6,CARD_3:def 3; then
	x in Seg 1 by A3,FINSEQ_1:38; then
	x = 1 by FINSEQ_1:2,TARSKI:def 1;
	hence y in [:[:X,X:],{1}:] by A5,A2,A6,A7,CARD_3:def 3;
	end;
	assume A8: y in [:[:X,X:],{1}:];
	1 in Seg 1 by FINSEQ_1:1; then
	A9: 1 in dom fs by A3,FINSEQ_1:38; then
	A10: 1 in dom disjoin fs by CARD_3:def 3;
	[:[:X,X:],{1}:] = (disjoin fs).1 by A2,A9,CARD_3:def 3; then
	[:[:X,X:],{1}:] in rng disjoin fs by A10,FUNCT_1:def 3;
	hence y in union rng disjoin fs by A8,TARSKI:def 4;
	end;
	thus free_magma(X,2) = union rng disjoin fs by A1,CARD_3:def 4
	.= [:[:X,X:],{1}:] by A4,TARSKI:2;
	end;

	theorem
	free_magma(X,3) = [:[:X,[:[:X,X:],{1}:]:],{1}:] \/
	[:[:[:[:X,X:],{1}:],X:],{2}:]
	proof
	set X1 = [:[:X,[:[:X,X:],{1}:]:],{1}:];
	set X2 = [:[:[:[:X,X:],{1}:],X:],{2}:];
	consider fs be FinSequence such that
	A1: len fs = 3-1 & (for p being Nat st p>=1 & p<=3-1 holds
	fs.p = [: (free_magma_seq X).p, (free_magma_seq X).(3-p) :] ) &
	(free_magma_seq X).3 = Union disjoin fs by Def13;
	A2: fs.1 = [: free_magma(X,1), free_magma(X,2) :] by A1
	.= [: free_magma(X,1), [:[:X,X:],{1}:] :] by Th18
	.= [:X,[:[:X,X:],{1}:]:] by Def13;
	A3: fs.2 = [: (free_magma_seq X).2, (free_magma_seq X).(3-2) :] by A1
	.= [: free_magma(X,2), X :] by Def13 .= [:[:[:X,X:],{1}:],X:] by Th18;
	A4: for y being object
	holds y in union rng disjoin fs iff y in X1 or y in X2
	proof
	let y be object;
	hereby
	assume y in union rng disjoin fs; then
	consider Y be set such that
	A5: y in Y & Y in rng disjoin fs by TARSKI:def 4;
	consider x be object such that
	A6: x in dom disjoin fs & Y = (disjoin fs).x by A5,FUNCT_1:def 3;
	A7: x in dom fs by A6,CARD_3:def 3; then
	x in {1,2} by A1,FINSEQ_1:2,def 3; then
	x = 1 or x = 2 by TARSKI:def 2;
	hence y in X1 or y in X2 by A2,A3,A5,A6,A7,CARD_3:def 3;
	end;
	assume A8: y in X1 or y in X2;
	1 in Seg 2 & 2 in Seg 2 by FINSEQ_1:1; then
	A9: 1 in dom fs & 2 in dom fs by A1,FINSEQ_1:def 3; then
	A10: 1 in dom disjoin fs & 2 in dom disjoin fs by CARD_3:def 3;
	X1 = (disjoin fs).1 & X2 = (disjoin fs).2 by A2,A3,A9,CARD_3:def 3; then
	X1 in rng disjoin fs & X2 in rng disjoin fs by A10,FUNCT_1:def 3;
	hence y in union rng disjoin fs by A8,TARSKI:def 4;
	end;
	thus free_magma(X,3) = union rng disjoin fs by A1,CARD_3:def 4
	.= [:[:X,[:[:X,X:],{1}:]:],{1}:] \/ [:[:[:[:X,X:],{1}:],X:],{2}:]
	by A4,XBOOLE_0:def 3;
	end;

	reserve x,y,Y for set;
	reserve n,m,p for Nat;

	theorem Th20:
	n >= 2 implies ex fs being FinSequence st len fs = n-1 &
	(for p st p>=1 & p<=n-1 holds
	fs.p = [: free_magma(X,p), free_magma(X,n-'p) :] ) &
	free_magma(X,n) = Union disjoin fs
	proof
	assume n >= 2; then
	consider fs be FinSequence such that
	A1: len fs = n-1 & (for p st p>=1 & p<=n-1 holds
	fs.p = [: (free_magma_seq X).p, (free_magma_seq X).(n-p) :] ) &
	(free_magma_seq X).n = Union disjoin fs by Def13;
	take fs;
	thus len fs = n-1 by A1;
	thus for p st p>=1 & p<=n-1
	holds fs.p = [: free_magma(X,p), free_magma(X,n-'p) :]
	proof
	let p;
	assume A2: p>=1 & p<=n-1; then
	-p <= -1 & -p >= -(n-1) by XREAL_1:24; then
	-p +n <= -1+n & -p+n >= -(n-1)+n by XREAL_1:6;
	then n-p=n-'p by XREAL_0:def 2;
	hence thesis by A2,A1;
	end;
	thus free_magma(X,n) = Union disjoin fs by A1;
	end;

	theorem Th21:
	n >= 2 & x in free_magma(X,n) implies ex p,m st x`2 = p & 1<=p & p<=n-1 &
	x`1`1 in free_magma(X,p) & x`1`2 in free_magma(X,m) & n = p + m &
	x in [:[:free_magma(X,p),free_magma(X,m):],{p}:]
	proof
	assume A1: n>=2;
	assume A2: x in free_magma(X,n);
	consider fs be FinSequence such that
	A3: len fs = n-1 and
	A4: (for p st p>=1 & p<=n-1 holds
	fs.p = [:(free_magma_seq X).p,(free_magma_seq X).(n-p) :] ) and
	A5: (free_magma_seq X).n = Union disjoin fs by A1,Def13;
	x in union rng disjoin fs by A2,A5,CARD_3:def 4; then
	consider Y be set such that
	A6: x in Y & Y in rng disjoin fs by TARSKI:def 4;
	consider p be object such that
	A7: p in dom disjoin fs & Y = (disjoin fs).p by A6,FUNCT_1:def 3;
	A8: p in dom fs by A7,CARD_3:def 3; then
	reconsider p as Nat;
	A9: p in Seg len fs by A8,FINSEQ_1:def 3; then
	A10: 1 <= p & p <= len fs by FINSEQ_1:1; then
	A11: fs.p = [:(free_magma_seq X).p,(free_magma_seq X).(n-p):] by A3,A4;
	then x in [:[:(free_magma_seq X).p,(free_magma_seq X).(n-p):],{p}:]
	by A6,A7,A8,CARD_3:def 3; then
	A12: x`1 in [:(free_magma_seq X).p,(free_magma_seq X).(n-p):] &
	x`2 in {p} by MCART_1:10;
	-p >= -(n-1) by A10,A3,XREAL_1:24; then
	-p+n >= -(n-1)+n by XREAL_1:7; then
	n-p in NAT by INT_1:3; then
	reconsider m = n-p as Nat;
	take p,m;
	thus thesis by A3,A9,A6,A11,A7,A8,A12,CARD_3:def 3,FINSEQ_1:1,MCART_1:10
	,TARSKI:def 1;
	end;

	theorem Th22:
	x in free_magma(X,n) & y in free_magma(X,m) implies
	[[x,y],n] in free_magma(X,n+m)
	proof
	assume A1: x in free_magma(X,n);
	assume A2: y in free_magma(X,m);
	per cases;
	suppose n=0 or m=0; hence thesis by Def13,A1,A2; end;
	suppose n<>0 & m<>0; then
	A3: n>=0+1 & m>=0+1 by NAT_1:13; then
	n+m>=1+1 by XREAL_1:7; then
	consider fs be FinSequence such that
	A4: len fs = (n+m)-1 & (for p st p>=1 & p<=(n+m)-1 holds
	fs.p = [: (free_magma_seq X).p,(free_magma_seq X).((n+m)-p) :] )
	& (free_magma_seq X).(n+m) = Union disjoin fs by Def13;
	1-1 <= m-1 by A3,XREAL_1:9; then
	A5: 0+n <= (m-1)+n by XREAL_1:7;
	then fs.n = [: (free_magma_seq X).n,(free_magma_seq X).((n+m)-n) :]
	by A4,A3
	.= [: (free_magma_seq X).n, (free_magma_seq X).m :];
	then A6: [x,y] in fs.n by A1,A2,ZFMISC_1:def 2;
	n in {n} by TARSKI:def 1; then
	A7: [[x,y],n] in [:fs.n, {n}:] by A6,ZFMISC_1:def 2;
	n in Seg len fs by A4,A3,A5,FINSEQ_1:1; then
	A8: n in dom fs by FINSEQ_1:def 3; then
	A9: (disjoin fs).n = [:fs.n,{n}:] by CARD_3:def 3;
	n in dom disjoin fs by A8,CARD_3:def 3; then
	[:fs.n,{n}:] in rng disjoin fs by A9,FUNCT_1:3; then
	[[x,y],n] in union rng disjoin fs by A7,TARSKI:def 4;
	hence [[x,y],n] in free_magma(X,n+m) by A4,CARD_3:def 4;
	end;
	end;

	theorem Th23:
	X c= Y implies free_magma(X,n) c= free_magma(Y,n)
	proof
	defpred P[Nat] means X c= Y implies free_magma(X,$1) c= free_magma(Y,$1);
	A1: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
	proof
	let k be Nat;
	assume A2: for n being Nat st n < k holds P[n];
	thus X c= Y implies free_magma(X,k) c= free_magma(Y,k)
	proof
	assume A3: X c= Y;
	k = 0 or k + 1 > 0+1 by XREAL_1:6; then
	k = 0 or k >= 1 by NAT_1:13; then
	k = 0 or k = 1 or k > 1 by XXREAL_0:1; then
	A4: k = 0 or k = 1 or k + 1 > 1+1 by XREAL_1:6;
	per cases by A4,NAT_1:13;
	suppose k=0; then free_magma(X,k) = {} & free_magma(Y,k) = {} by Def13;
	hence free_magma(X,k) c= free_magma(Y,k);
	end;
	suppose k=1; then free_magma(X,k) = X & free_magma(Y,k) = Y by Def13;
	hence free_magma(X,k) c= free_magma(Y,k) by A3;
	end;
	suppose A5: k>=2;
	for x being object st x in free_magma(X,k) holds x in free_magma(Y,k)
	proof
	let x be object;
	assume x in free_magma(X,k); then
	consider p,m be Nat such that
	A6: x`2 = p & 1<=p & p<=k-1 &
	x`1`1 in free_magma(X,p) & x`1`2 in free_magma(X,m) & k = p + m &
	x in [:[:free_magma(X,p),free_magma(X,m):],{p}:] by A5,Th21;
	consider fs be FinSequence such that
	A7: len fs = k-1 & (for p being Nat st p>=1 & p<=k-1 holds
	fs.p = [: free_magma(Y,p), free_magma(Y,k-'p) :]) &
	free_magma(Y,k) = Union disjoin fs by A5,Th20;
	A8: fs.p = [: free_magma(Y,p), free_magma(Y,k-'p) :] by A6,A7;
	A9: x`1 in [:free_magma(X,p),free_magma(X,m):] & x`2 in {p}
	by A6,MCART_1:10;
	A10: x = [x`1,x`2] by A6,MCART_1:21;
	A11: x`1 = [x`1`1,x`1`2] by A9,MCART_1:21;
	p+1 <= k-1+1 by A6,XREAL_1:7; then
	A12: p<k by NAT_1:13; then
	A13: free_magma(X,p) c= free_magma(Y,p) by A2,A3;
	p-p<k-p by A12,XREAL_1:14; then
	A14: k-'p = m by A6,XREAL_0:def 2;
	p+m>0+m by A6,XREAL_1:8; then
	free_magma(X,m) c= free_magma(Y,k-'p) by A6,A2,A3,A14; then
	x`1 in [:free_magma(Y,p),free_magma(Y,k-'p):]
	by A6,A11,A13,ZFMISC_1:def 2; then
	A15: x in [:fs.p,{p}:] by A8,A10,A9,ZFMISC_1:def 2;
	p in Seg len fs by A6,A7,FINSEQ_1:1; then
	A16: p in dom fs by FINSEQ_1:def 3; then
	A17: (disjoin fs).p = [:fs.p,{p}:] by CARD_3:def 3;
	p in dom disjoin fs by A16,CARD_3:def 3; then
	[:fs.p,{p}:] in rng disjoin fs by A17,FUNCT_1:3; then
	x in union rng disjoin fs by A15,TARSKI:def 4;
	hence x in free_magma(Y,k) by A7,CARD_3:def 4;
	end;
	hence free_magma(X,k) c= free_magma(Y,k);
	end;
	end;
	end;
	for k being Nat holds P[k] from NAT_1:sch 4(A1);
	hence thesis;
	end;

	definition
	let X be set;
	func free_magma_carrier X -> set
	equals Union disjoin((free_magma_seq X)\|NATPLUS);
	correctness;
	end;

	Lm1: n>0 implies [:free_magma(X,n),{n}:] c= free_magma_carrier X
	proof
	assume A1: n > 0;
	let x be object;
	assume A2: x in [:free_magma(X,n),{n}:];
	n in NAT by ORDINAL1:def 12; then
	A3: n in dom free_magma_seq X by FUNCT_2:def 1;
	n in NATPLUS by A1,NAT_LAT:def 6; then
	A4: n in dom ((free_magma_seq X)\|NATPLUS) by A3,RELAT_1:57; then
	A5: (disjoin((free_magma_seq X)\|NATPLUS)).n
	= [:((free_magma_seq X)\|NATPLUS).n,{n}:] by CARD_3:def 3
	.= [:(free_magma_seq X).n,{n}:] by A4,FUNCT_1:47;
	n in dom disjoin((free_magma_seq X)\|NATPLUS) by A4,CARD_3:def 3; then
	[:free_magma(X,n),{n}:] in rng disjoin((free_magma_seq X)\|NATPLUS)
	by A5,FUNCT_1:3;
	then
	x in union rng disjoin((free_magma_seq X)\|NATPLUS) by A2,TARSKI:def 4;
	hence x in free_magma_carrier X by CARD_3:def 4;
	end;

	theorem Th24:
	X = {} iff free_magma_carrier X = {}
	proof
	hereby
	assume A1: X = {};
	defpred P[Nat] means (free_magma_seq X).$1 = {};
	A2: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
	proof
	let k be Nat;
	assume A3: for n being Nat st n < k holds P[n];
	k = 0 or k + 1 > 0+1 by XREAL_1:6; then
	k = 0 or k >= 1 by NAT_1:13; then
	k = 0 or k = 1 or k > 1 by XXREAL_0:1; then
	A4: k = 0 or k = 1 or k + 1 > 1+1 by XREAL_1:6;
	per cases by A4,NAT_1:13;
	suppose k=0; hence P[k] by Def13; end;
	suppose k=1; hence P[k] by A1,Def13; end;
	suppose k>=2; then
	consider fs be FinSequence such that
	A5: len fs = k-1 & (for p being Nat st p>=1 & p<=k-1 holds
	fs.p = [: (free_magma_seq X).p, (free_magma_seq X).(k-p) :] ) &
	(free_magma_seq X).k = Union disjoin fs by Def13;
	for y being set st y in rng disjoin fs holds y c= {}
	proof
	let y be set;
	assume y in rng disjoin fs; then
	consider p be object such that
	A6: p in dom disjoin fs & y = (disjoin fs).p by FUNCT_1:def 3;
	A7: p in dom fs by A6,CARD_3:def 3; then
	A8: p in Seg len fs by FINSEQ_1:def 3;
	reconsider p as Nat by A7;
	A9: p >= 1 & p <= k-1 by A5,A8,FINSEQ_1:1; then
	p+1 <= k-1+1 by XREAL_1:7; then
	p < k by NAT_1:13; then
	A10: (free_magma_seq X).p = {} by A3;
	A11: fs.p = [:(free_magma_seq X).p,(free_magma_seq X).(k-p):]
	by A5,A9
	.= {} by A10,ZFMISC_1:90;
	(disjoin fs).p = [:fs.p,{p}:] by A7,CARD_3:def 3
	.= {} by A11,ZFMISC_1:90;
	hence y c= {} by A6;
	end; then
	union rng disjoin fs c= {} by ZFMISC_1:76;
	hence P[k] by A5,CARD_3:def 4;
	end;
	end;
	A12: for n being Nat holds P[n] from NAT_1:sch 4(A2);
	for Y being set st Y in rng disjoin((free_magma_seq X)\|NATPLUS)
	holds Y c= {}
	proof
	let Y be set;
	assume Y in rng disjoin((free_magma_seq X)\|NATPLUS); then
	consider n be object such that
	A13: n in dom disjoin((free_magma_seq X)\|NATPLUS) &
	Y = (disjoin((free_magma_seq X)\|NATPLUS)).n by FUNCT_1:def 3;
	A14: n in dom((free_magma_seq X)\|NATPLUS) by A13,CARD_3:def 3; then
	reconsider n as Nat;
	A15: n in dom ((free_magma_seq X)\|NATPLUS) by A13,CARD_3:def 3;
	(disjoin((free_magma_seq X)\|NATPLUS)).n
	= [:((free_magma_seq X)\|NATPLUS).n,{n}:] by A14,CARD_3:def 3
	.= [:(free_magma_seq X).n,{n}:] by A15,FUNCT_1:47
	.= [:{},{n}:] by A12 .= {} by ZFMISC_1:90;
	hence Y c= {} by A13;
	end; then
	union rng disjoin((free_magma_seq X)\|NATPLUS) c= {} by ZFMISC_1:76;
	hence free_magma_carrier X = {} by CARD_3:def 4;
	end;
	assume A16: free_magma_carrier X = {};
	[:free_magma(X,1),{1}:] c= free_magma_carrier X by Lm1;
	hence X = {} by A16;
	end;

	registration
	let X be empty set;
	cluster free_magma_carrier X -> empty;
	correctness by Th24;
	end;

	registration
	let X be non empty set;
	cluster free_magma_carrier X -> non empty;
	correctness by Th24;
	let w be Element of free_magma_carrier X;
	cluster w`2 -> non zero natural for number;
	correctness
	proof
	w in free_magma_carrier X; then
	w in union rng disjoin((free_magma_seq X)\|NATPLUS) by CARD_3:def 4; then
	consider Y be set such that
	A1: w in Y & Y in rng disjoin((free_magma_seq X)\|NATPLUS) by TARSKI:def 4;
	consider n be object such that
	A2: n in dom disjoin((free_magma_seq X)\|NATPLUS) &
	Y = disjoin((free_magma_seq X)\|NATPLUS).n by A1,FUNCT_1:def 3;
	A3: n in dom((free_magma_seq X)\|NATPLUS) by A2,CARD_3:def 3; then
	n in NATPLUS by RELAT_1:57; then
	reconsider n as non zero Nat;
	w in [:((free_magma_seq X)\|NATPLUS).n,{n}:] by A2,A1,A3,CARD_3:def 3; then
	w`2 in {n} by MCART_1:10;
	hence thesis by TARSKI:def 1;
	end;
	end;

	theorem Th25:
	for X being non empty set, w being Element of free_magma_carrier X
	holds w in [:free_magma(X,w`2),{w`2}:]
	proof
	let X be non empty set;
	let w be Element of free_magma_carrier X;
	w in free_magma_carrier X; then
	w in union rng disjoin((free_magma_seq X)\|NATPLUS) by CARD_3:def 4; then
	consider Y be set such that
	A1: w in Y & Y in rng disjoin((free_magma_seq X)\|NATPLUS) by TARSKI:def 4;
	consider n be object such that
	A2: n in dom disjoin((free_magma_seq X)\|NATPLUS) &
	Y = disjoin((free_magma_seq X)\|NATPLUS).n by A1,FUNCT_1:def 3;
	A3: n in dom((free_magma_seq X)\|NATPLUS) by A2,CARD_3:def 3; then
	A4: ((free_magma_seq X)\|NATPLUS).n = (free_magma_seq X).n by FUNCT_1:47;
	reconsider n as Nat by A3;
	w in [:((free_magma_seq X)\|NATPLUS).n,{n}:] by A2,A1,A3,CARD_3:def 3; then
	w`2 in {n} by MCART_1:10; then
	w`2 = n by TARSKI:def 1;
	hence w in [:free_magma(X,w`2),{w`2}:] by A4,A2,A1,A3,CARD_3:def 3;
	end;

	theorem Th26:
	for X being non empty set, v,w being Element of free_magma_carrier X
	holds [[[v`1,w`1],v`2],v`2+w`2] is Element of free_magma_carrier X
	proof
	let X be non empty set;
	let v,w be Element of free_magma_carrier X;
	v in [:free_magma(X,v`2),{v`2}:] by Th25; then
	A1: v`1 in free_magma(X,v`2) by MCART_1:10;
	w in [:free_magma(X,w`2),{w`2}:] by Th25; then
	w`1 in free_magma(X,w`2) by MCART_1:10; then
	A2: [[v`1,w`1],v`2] in free_magma(X,v`2+w`2) by A1,Th22;
	A3: v`2 + w`2 in {v`2 + w`2} by TARSKI:def 1;
	set z = [[[v`1,w`1],v`2],v`2+w`2];
	A4: z`1 in free_magma(X,v`2+w`2) by A2;
	z`2 in {v`2 + w`2} by A3; then
	A5: z in [:free_magma(X,v`2+w`2),{v`2+w`2}:] by A4,ZFMISC_1:def 2;
	[:free_magma(X,v`2 + w`2),{v`2 + w`2}:] c= free_magma_carrier X by Lm1;
	hence thesis by A5;
	end;

	theorem Th27:
	X c= Y implies free_magma_carrier X c= free_magma_carrier Y
	proof
	assume A1: X c= Y;
	per cases;
	suppose X = {}; then
	free_magma_carrier X = {};
	hence free_magma_carrier X c= free_magma_carrier Y;
	end;
	suppose A2: X <> {};
	let x be object;
	assume A3: x in free_magma_carrier X;
	reconsider X9=X as non empty set by A2;
	reconsider w=x as Element of free_magma_carrier X9 by A3;
	A4: w in [:free_magma(X9,w`2),{w`2}:] by Th25; then
	A5: w`1 in free_magma(X9,w`2) & w`2 in {w`2} by MCART_1:10;
	reconsider Y9=Y as non empty set by A2,A1;
	A6: free_magma(X9,w`2) c= free_magma(Y9,w`2) by A1,Th23;
	w = [w`1,w`2] by A4,MCART_1:21; then
	A7: w in [:free_magma(Y9,w`2),{w`2}:] by A6,A5,ZFMISC_1:def 2;
	[:free_magma(Y9,w`2),{w`2}:] c= free_magma_carrier Y9 by Lm1;
	hence x in free_magma_carrier Y by A7;
	end;
	end;

	theorem
	n>0 implies [:free_magma(X,n),{n}:] c= free_magma_carrier X by Lm1;

	definition
	let X be set;
	func free_magma_mult X -> BinOp of free_magma_carrier X means :Def16:
	for v,w being Element of free_magma_carrier X, n,m st n = v`2 & m = w`2
	holds it.(v,w) = [[[v`1,w`1],v`2],n+m]
	if X is non empty otherwise it = {};
	correctness
	proof
	A1: X is non empty implies ex f being BinOp of free_magma_carrier X st
	for v,w being Element of free_magma_carrier X, n,m st n = v`2 & m = w`2
	holds f.(v,w) = [[[v`1,w`1],v`2],n+m]
	proof
	assume A2: X is non empty;
	defpred P[set,set,set] means for n,m st n=$1`2 & m=$2`2
	holds $3 = [[[$1`1,$2`1],$1`2],n+m];
	reconsider Y = free_magma_carrier X as non empty set by A2;
	A3: for x being Element of Y for y being Element of Y
	ex z being Element of Y st P[x,y,z]
	proof
	let x be Element of Y;
	let y be Element of Y;
	reconsider X9=X as non empty set by A2;
	reconsider v=x as Element of free_magma_carrier X9;
	reconsider w=y as Element of free_magma_carrier X9;
	reconsider z=[[[v`1,w`1],v`2],v`2+w`2] as Element of Y by Th26;
	take z;
	thus thesis;
	end;
	consider f be Function of [:Y,Y:],Y such that
	A4: for x being Element of Y for y being Element of Y
	holds P[x,y,f.(x,y)] from BINOP_1:sch 3(A3);
	reconsider f as BinOp of free_magma_carrier X;
	take f;
	thus thesis by A4;
	end;
	A5: X is empty implies ex f being BinOp of free_magma_carrier X st f = {}
	proof
	assume A6: X is empty; then
	A7: free_magma_carrier X = {};
	{} c= [:{} qua set,{} qua set:]; then
	reconsider f = {} as Relation of [:{} qua set,{} qua set:],{}
	by ZFMISC_1:90;
	([:{} qua set,{} qua set:] = {} implies {}={}) &
	dom f = [:{} qua set,{} qua set:] by ZFMISC_1:90; then
	reconsider f = {} as BinOp of {} by FUNCT_2:def 1;
	for v,w being Element of free_magma_carrier X, n,m st n = v`2 & m = w`2 &
	v in free_magma_carrier X & w in free_magma_carrier X holds
	f.(v,w) = [[[v`1,w`1],v`2],n+m] by A6;
	hence thesis by A7;
	end;
	now
	let f1, f2 be BinOp of free_magma_carrier X;
	thus X is non empty & ( for v, w being Element of free_magma_carrier X,
	n,m st n = v`2 & m = w`2 holds f1.(v,w) = [[[v`1,w`1],v`2],n+m])
	& (for v, w being Element of free_magma_carrier X,
	n,m st n = v`2 & m = w`2 holds f2.(v,w) = [[[v`1,w`1],v`2],n+m])
	implies f1 = f2
	proof
	assume A8: X is non empty;
	assume A9: for v, w being Element of free_magma_carrier X,
	n,m st n = v`2 & m = w`2 holds f1.(v,w) = [[[v`1,w`1],v`2],n+m];
	assume A10: for v, w being Element of free_magma_carrier X,
	n,m st n = v`2 & m = w`2 holds f2.(v,w) = [[[v`1,w`1],v`2],n+m];
	for v,w being Element of free_magma_carrier X holds f1.(v,w) = f2.(v,w)
	proof
	let v,w be Element of free_magma_carrier X;
	set n = v`2, m = w`2;
	reconsider n,m as Nat by A8;
	thus f1.(v,w) = [[[v`1,w`1],v`2],n+m] by A9 .= f2.(v,w) by A10;
	end;
	hence f1 = f2 by BINOP_1:2;
	end;
	assume X is empty & f1 = {} & f2 = {}; hence thesis;
	end;
	hence thesis by A1,A5;
	end;
	end;

	:: Ch I ?7.1 Algebra I Bourbaki

	definition
	let X be set;
	func free_magma X -> multMagma equals
	multMagma(# free_magma_carrier X, free_magma_mult X #);
	correctness;
	end;

	registration
	let X be set;
	cluster free_magma X -> strict;
	correctness;
	end;

	registration
	let X be empty set;
	cluster free_magma X -> empty;
	correctness;
	end;

	registration
	let X be non empty set;
	cluster free_magma X -> non empty;
	correctness;
	let w be Element of free_magma X;
	cluster w`2 -> non zero natural for number;
	correctness;
	end;

	theorem
	for X being set, S being Subset of X
	holds free_magma S is multSubmagma of free_magma X
	proof
	let X be set;
	let S be Subset of X;
	A1: the carrier of free_magma S c= the carrier of free_magma X by Th27;
	reconsider A = the carrier of free_magma S as set;
	A2: (the multF of free_magma X) \| [: A, A :]
	= (the multF of free_magma X)\|\|the carrier of free_magma S by REALSET1:def 2;
	per cases;
	suppose A3: S is empty;
	then A4: the carrier of free_magma S = {};
	the multF of free_magma S = (the multF of free_magma X) \| {} by A3
	.= (the multF of free_magma X) \| [: A, A :] by A4,ZFMISC_1:90;
	hence thesis by A2,A1,Def9;
	end;
	suppose A5: S is not empty;
	then A6: dom the multF of free_magma S = [:A,A:] by FUNCT_2:def 1;
	A7: X is non empty by A5;
	[:A,A:] c= [: free_magma_carrier X, free_magma_carrier X:]
	by A1,ZFMISC_1:96; then
	[:A,A:] c= dom the multF of free_magma X by A7,FUNCT_2:def 1;
	then A8: dom the multF of free_magma S
	= dom((the multF of free_magma X)\|\|the carrier of free_magma S)
	by A6,A2,RELAT_1:62;
	for z being object st z in dom the multF of free_magma S holds
	(the multF of free_magma S).z
	=((the multF of free_magma X)\|\|the carrier of free_magma S).z
	proof
	let z be object;
	assume A9: z in dom the multF of free_magma S;
	then consider x,y being object such that
	A10: x in A & y in A & z=[x,y] by ZFMISC_1:def 2;
	reconsider x,y as Element of free_magma_carrier S by A10;
	reconsider n=x`2,m=y`2 as Nat by A5;
	reconsider x9=x,y9=y as Element of free_magma_carrier X by A10,A1;
	(the multF of free_magma S).z
	= (the multF of free_magma S).(x,y) by A10,BINOP_1:def 1
	.= [[[x`1,y`1],x`2],n+m] by A5,Def16
	.= (free_magma_mult X).(x9,y9) by A7,Def16
	.= (the multF of free_magma X).z by A10,BINOP_1:def 1
	.= ((the multF of free_magma X)\|[:A,A:]).z by A9,FUNCT_1:49;
	hence thesis by REALSET1:def 2;
	end;
	then the multF of free_magma S
	= (the multF of free_magma X)\|\|the carrier of free_magma S
	by A8,FUNCT_1:2;
	hence free_magma S is multSubmagma of free_magma X by A1,Def9;
	end;
	end;

	definition
	let X be set;
	let w be Element of free_magma X;
	func length w -> Nat equals :Def18:
	w`2 if X is non empty otherwise 0;
	correctness;
	end;

	theorem Th30:
	X = {w`1 where w is Element of free_magma X: length w = 1}
	proof
	for x being object holds x in X iff
	x in {w`1 where w is Element of free_magma X: length w = 1}
	proof
	let x be object;
	hereby
	assume A1: x in X; then
	A2: x in free_magma(X,1) by Def13;
	1 in {1} by TARSKI:def 1; then
	A3: [x,1] in [:free_magma(X,1),{1}:] by A2,ZFMISC_1:def 2;
	[:free_magma(X,1),{1}:] c= free_magma_carrier X by Lm1; then
	reconsider w9 = [x,1] as Element of free_magma X by A3;
	1 = [x,1]`2; then
	A4: length w9 = 1 by A1,Def18;
	x = [x,1]`1;
	hence x in {w`1 where w is Element of free_magma X: length w = 1} by A4;
	end;
	assume x in {w`1 where w is Element of free_magma X: length w = 1}; then
	consider w be Element of free_magma X such that
	A5: x = w`1 & length w = 1;
	A6: w`2 = 1 by A5,Def18;
	per cases;
	suppose X is non empty; then
	w in [:free_magma(X,1),{1}:] by A6,Th25; then
	w in [:X,{1}:] by Def13;
	hence x in X by A5,MCART_1:10;
	end;
	suppose X is empty; hence thesis by A5,Def18; end;
	end;
	hence thesis by TARSKI:2;
	end;

	reserve v,v1,v2,w,w1,w2 for Element of free_magma X;

	theorem Th31:
	X is non empty implies v*w = [[[v`1,w`1],v`2],length v + length w]
	proof
	assume A1: X is non empty; then
	length v = v`2 & length w = w`2 by Def18;
	hence thesis by A1,Def16;
	end;

	theorem Th32:
	X is non empty implies v = [v`1,v`2] & length v >= 1
	proof
	assume X is non empty; then
	reconsider X9=X as non empty set;
	reconsider v9=v as Element of free_magma X9;
	v9 in [:free_magma(X,v9`2),{v9`2}:] by Th25; then
	ex x,y being object st x in free_magma(X,v9`2) &
	y in {v9`2} & v9=[x,y] by ZFMISC_1:def 2;
	hence v = [v`1,v`2];
	reconsider v99=v9 as Element of free_magma_carrier X9;
	v99`2 > 0; then
	length v9 > 0 by Def18; then
	length v9+1 > 0+1 by XREAL_1:6;
	hence thesis by NAT_1:13;
	end;

	theorem
	length(v*w) = length v + length w
	proof
	set vw = v*w;
	per cases;
	suppose A1: X is non empty; then
	v*w = [[[v`1,w`1],v`2],length v + length w] by Th31;
	hence length(v*w) = [[[v`1,w`1],v`2],length v + length w]`2 by A1,Def18
	.= length v + length w;
	end;
	suppose A2: X is empty;
	hence length(v*w) = 0 by Def18
	.= length v + 0 by A2,Def18
	.= length v + length w by A2,Def18;
	end;
	end;

	theorem Th34:
	length w >= 2 implies
	ex w1,w2 st w = w1*w2 & length w1 < length w & length w2 < length w
	proof
	assume A1: length w >= 2; then
	reconsider X9=X as non empty set by Def18;
	reconsider w9=w as Element of free_magma X9;
	A2: w9 in [:free_magma(X,w9`2),{w9`2}:] by Th25;
	set n = length w;
	A3: n = w9`2 by Def18;
	consider fs be FinSequence such that
	A4: len fs = n-1 and
	A5: (for p being Nat st p>=1 & p<=n-1 holds
	fs.p = [:(free_magma_seq X).p,(free_magma_seq X).(n-p) :] ) and
	A6: (free_magma_seq X).n = Union disjoin fs by A1,Def13;
	w9`1 in (free_magma_seq X).n by A3,A2,MCART_1:10; then
	w9`1 in union rng disjoin fs by A6,CARD_3:def 4; then
	consider Y be set such that
	A7: w9`1 in Y & Y in rng disjoin fs by TARSKI:def 4;
	consider p be object such that
	A8: p in dom disjoin fs & Y = (disjoin fs).p by A7,FUNCT_1:def 3;
	A9: p in dom fs by A8,CARD_3:def 3; then
	reconsider p as Nat;
	A10: p in Seg len fs by A9,FINSEQ_1:def 3; then
	A11: 1 <= p & p <= len fs by FINSEQ_1:1; then
	fs.p = [:(free_magma_seq X).p,(free_magma_seq X).(n-p):] by A4,A5;
	then A12: w9`1 in [:[:(free_magma_seq X).p,(free_magma_seq X).(n-p):],{p}:]
	by A7,A8,A9,CARD_3:def 3; then
	A13: w9`1`1 in [:(free_magma_seq X).p,(free_magma_seq X).(n-p):] &
	w9`1`2 in {p} by MCART_1:10; then
	A14: w9`1`1`1 in (free_magma_seq X).p & w9`1`1`2 in (free_magma_seq X).(n-p)
	by MCART_1:10;
	-p >= -(n-1) by A11,A4,XREAL_1:24; then
	A15: -p+n >= -(n-1)+n by XREAL_1:7; then
	n-p in NAT by INT_1:3; then
	reconsider m = n-p as Nat;
	set w19 = [w9`1`1`1,p];
	set w29 = [w9`1`1`2,m];
	p in {p} by TARSKI:def 1; then
	A16: w19 in [: free_magma(X,p),{p}:] by A14,ZFMISC_1:def 2;
	m in {m} by TARSKI:def 1; then
	A17: w29 in [: free_magma(X,m),{m}:] by A14,ZFMISC_1:def 2;
	[: free_magma(X,p),{p}:] c= free_magma_carrier X by A11,Lm1; then
	reconsider w19 as Element of free_magma_carrier X by A16;
	[: free_magma(X,m),{m}:] c= free_magma_carrier X by A15,Lm1; then
	reconsider w29 as Element of free_magma_carrier X by A17;
	reconsider w1=w19,w2=w29 as Element of free_magma X;
	A18: length w1 = [w9`1`1`1,p]`2 by Def18 .= p;
	A19: length w2 = [w9`1`1`2,m]`2 by Def18 .= m;
	ex x,y being object
	st x in [:(free_magma_seq X).p,(free_magma_seq X).(n-p):] &
	y in {p} & w9`1=[x,y] by A12,ZFMISC_1:def 2; then
	A20: w9`1 = [w9`1`1,w9`1`2] .= [w9`1`1,p] by A13,TARSKI:def 1;
	A21: ex x,y being object st x in (free_magma_seq X).p &
	y in (free_magma_seq X).(n-p) & w9`1`1=[x,y] by A13,ZFMISC_1:def 2;
	take w1,w2;
	ex x,y being object st x in free_magma(X,w9`2) &
	y in {w9`2} & w9=[x,y] by A2,ZFMISC_1:def 2;
	hence w = [w9`1, w9`2]
	.= [[w9`1`1,p],n] by A20,Def18
	.= [[w9`1`1,w1`2],length w1 + length w2] by A18,A19
	.= [[[w9`1`1`1,w9`1`1`2],w1`2],length w1 + length w2] by A21
	.= [[[w9`1`1`1,w2`1],w1`2],length w1 + length w2]
	.= [[[w1`1,w2`1],w1`2],length w1 + length w2]
	.= w1*w2 by Th31;
	p <= (n-1) by A10,A4,FINSEQ_1:1; then
	p+1 <= (n-1)+1 by XREAL_1:7;
	hence length w1 < length w by A18,NAT_1:13;
	-1 >= -p by A11,XREAL_1:24; then
	-1+(n+1) >= -p+(n+1) by XREAL_1:7; then
	n >= m+1;
	hence length w2 < length w by A19,NAT_1:13;
	end;

	theorem
	v1v2 = w1w2 implies v1 = w1 & v2 = w2
	proof
	assume A1: v1v2 = w1w2;
	per cases;
	suppose A2: X is non empty; then
	v1*v2 = [[[v1`1,v2`1],v1`2],length v1 + length v2] &
	w1*w2 = [[[w1`1,w2`1],w1`2],length w1 + length w2] by Th31; then
	A3: [[v1`1,v2`1],v1`2] = [[w1`1,w2`1],w1`2] &
	length v1 + length v2 = length w1 + length w2 by A1,XTUPLE_0:1; then
	A4: [v1`1,v2`1] = [w1`1,w2`1] & v1`2 = w1`2 by XTUPLE_0:1;
	length v1 = v1`2 by A2,Def18 .= length w1 by A2,A4,Def18; then
	v2`2 = length w2 by A2,A3,Def18; then
	A5: v2`2 = w2`2 by A2,Def18;
	thus v1 = [v1`1,v1`2] by A2,Th32 .= [w1`1,w1`2] by A4,XTUPLE_0:1
	.= w1 by A2,Th32;
	thus v2 = [v2`1,v2`2] by A2,Th32 .= [w2`1,w2`2] by A5,A4,XTUPLE_0:1
	.= w2 by A2,Th32;
	end;
	suppose X is empty; then
	v1 = {} & w1 = {} & v2 = {} & w2 = {} by SUBSET_1:def 1;
	hence thesis;
	end;
	end;

	definition
	let X be set;
	let n be Nat;
	func canon_image(X,n) -> Function of free_magma(X,n),free_magma X means
	:Def19:
	for x st x in dom it holds it.x = [x,n] if n > 0 otherwise it = {};
	correctness
	proof
	A1: n > 0 implies ex f being Function of free_magma(X,n),free_magma X st
	for x st x in dom f holds f.x = [x,n]
	proof
	assume A2: n > 0;
	deffunc F(object) = [$1,n];
	A3: for x being object st x in free_magma(X,n)
	holds F(x) in the carrier of free_magma X
	proof
	let x be object;
	assume A4: x in free_magma(X,n);
	n in {n} by TARSKI:def 1; then
	A5: F(x) in [:free_magma(X,n),{n}:] by A4,ZFMISC_1:def 2;
	[:free_magma(X,n),{n}:] c= free_magma_carrier X by A2,Lm1;
	hence F(x) in the carrier of free_magma X by A5;
	end;
	consider f be Function of free_magma(X,n),the carrier of free_magma X
	such that
	A6: for x being object st x in free_magma(X,n) holds f.x = F(x)
	from FUNCT_2:sch 2(A3);
	take f;
	let x;
	assume x in dom f;
	hence f.x = [x,n] by A6;
	end;
	A7: not n > 0 implies
	ex f being Function of free_magma(X,n),free_magma X st f = {}
	proof
	assume not n > 0; then
	n = 0; then
	A8: free_magma(X,n) = {} by Def13;
	set f = {};
	A9: dom f = {};
	rng f c= the carrier of free_magma X; then
	reconsider f as Function of free_magma(X,n),free_magma X
	by A8,A9,FUNCT_2:2;
	take f;
	thus f = {};
	end;
	for f1,f2 being Function of free_magma(X,n),free_magma X holds n > 0 &
	(for x st x in dom f1 holds f1.x = [x,n] ) &
	(for x st x in dom f2 holds f2.x = [x,n] ) implies f1 = f2
	proof
	let f1,f2 be Function of free_magma(X,n),free_magma X;
	assume n > 0;
	assume A10: for x st x in dom f1 holds f1.x = [x,n];
	assume A11: for x st x in dom f2 holds f2.x = [x,n];
	per cases;
	suppose X is empty; hence thesis; end;
	suppose A12: X is non empty; then
	A13: dom f1 = free_magma(X,n) by FUNCT_2:def 1
	.= dom f2 by A12,FUNCT_2:def 1;
	for x being object st x in dom f1 holds f1.x = f2.x
	proof
	let x be object;
	assume A14: x in dom f1;
	hence f1.x = [x,n] by A10 .= f2.x by A11,A13,A14;
	end;
	hence thesis by A13,FUNCT_1:2;
	end;
	end;
	hence thesis by A1,A7;
	end;
	end;

	Lm2:
	canon_image(X,n) is one-to-one
	proof
	for x1,x2 being object st x1 in dom canon_image(X,n) &
	x2 in dom canon_image(X,n) & canon_image(X,n).x1 = canon_image(X,n).x2
	holds x1 = x2
	proof
	let x1,x2 be object;
	assume A1: x1 in dom canon_image(X,n) & x2 in dom canon_image(X,n);
	assume A2: canon_image(X,n).x1 = canon_image(X,n).x2;
	per cases;
	suppose n>0; then
	canon_image(X,n).x1 = [x1,n] & canon_image(X,n).x2 = [x2,n] by A1,Def19;
	hence x1 = x2 by A2,XTUPLE_0:1;
	end;
	suppose not n>0; then
	canon_image(X,n) = {} by Def19;
	hence thesis by A1;
	end;
	end;
	hence canon_image(X,n) is one-to-one by FUNCT_1:def 4;
	end;

	registration
	let X be set;
	let n be Nat;
	cluster canon_image(X,n) -> one-to-one;
	correctness by Lm2;
	end;

	reserve X,Y,Z for non empty set;

	Lm3: dom canon_image(X,1) = X &
	for x being set st x in X holds canon_image(X,1).x = [x,1]
	proof
	dom canon_image(X,1) = free_magma(X,1) by FUNCT_2:def 1;
	hence dom canon_image(X,1) = X by Def13;
	hence
	for x being set st x in X holds canon_image(X,1).x = [x,1] by Def19;
	end;

	theorem Th36:
	for A being Subset of free_magma X st A = canon_image(X,1) .: X
	holds free_magma X = the_submagma_generated_by A
	proof
	let A be Subset of free_magma X;
	set N = the_submagma_generated_by A;
	assume A1: A = canon_image(X,1) .: X;
	per cases;
	suppose A2: A is empty;
	X is empty
	proof
	assume X is non empty;
	consider x being object such that
	A3: x in X by XBOOLE_0:def 1;
	x in dom canon_image(X,1) by Lm3,A3; then
	canon_image(X,1).x in canon_image(X,1) .: X by A3,FUNCT_1:def 6;
	hence contradiction by A2,A1;
	end;
	hence thesis;
	end;
	suppose A4: A is not empty;
	A5: the carrier of N c= the carrier of free_magma X by Def9;
	for x being object
	st x in the carrier of free_magma X holds x in the carrier of N
	proof
	let x be object;
	assume A6: x in the carrier of free_magma X;
	defpred P[Nat] means for v being Element of free_magma X st
	length v = $1 holds v in the carrier of N;
	A7: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
	proof
	let k be Nat;
	assume A8: for n being Nat st n < k holds P[n];
	k = 0 or k + 1 > 0+1 by XREAL_1:6; then
	k = 0 or k >= 1 by NAT_1:13; then
	k = 0 or k = 1 or k > 1 by XXREAL_0:1; then
	A9: k = 0 or k = 1 or k + 1 > 1+1 by XREAL_1:6;
	per cases by A9,NAT_1:13;
	suppose k = 0; hence P[k] by Th32; end;
	suppose A10: k = 1;
	for v being Element of free_magma X st length v = 1
	holds v in the carrier of N
	proof
	let v be Element of free_magma X;
	assume A11: length v = 1;
	A12: v = [v`1,v`2] by Th32
	.= [v`1,1] by A11,Def18;
	v`1 in {w`1 where w is Element of free_magma X: length w = 1}
	by A11; then
	v`1 in X by Th30;
	then v`1 in dom canon_image(X,1) & v`1 in X &
	v = canon_image(X,1).(v`1) by A12,Lm3; then
	A13: v in A by A1,FUNCT_1:def 6;
	A c= the carrier of N by Def12;
	hence thesis by A13;
	end;
	hence P[k] by A10;
	end;
	suppose A14: k >= 2;
	for v being Element of free_magma X st length v = k
	holds v in the carrier of N
	proof
	let v be Element of free_magma X;
	assume A15: length v = k; then
	consider v1,v2 be Element of free_magma X such that
	A16: v = v1*v2 & length v1 < length v & length v2 < length v
	by A14,Th34;
	A17: v1 in the carrier of N by A8,A15,A16;
	reconsider v19=v1 as Element of N by A8,A15,A16;
	A18: v2 in the carrier of N by A8,A15,A16;
	reconsider v29=v2 as Element of N by A8,A15,A16;
	N is non empty by A4,Th14; then
	A19: the carrier of N <> {};
	A20: [v1,v2] in [:the carrier of N,the carrier of N:]
	by A17,A18,ZFMISC_1:87;
	v19*v29 = (the multF of N).[v19,v29] by BINOP_1:def 1
	.= ((the multF of free_magma X)\|\|the carrier of N).[v1,v2] by Def9
	.= ((the multF of free_magma X)\|
	[:the carrier of N,the carrier of N:]).[v1,v2] by REALSET1:def 2
	.= (the multF of free_magma X).[v1,v2] by A20,FUNCT_1:49
	.= v1*v2 by BINOP_1:def 1;
	hence v in the carrier of N by A16,A19;
	end;
	hence P[k];
	end;
	end;
	A21: for n being Nat holds P[n] from NAT_1:sch 4(A7);
	reconsider v = x as Element of free_magma X by A6;
	reconsider k = length v as Nat;
	P[k] by A21;
	hence x in the carrier of N;
	end; then
	the carrier of free_magma X c= the carrier of N; then
	the carrier of free_magma X = the carrier of N by A5,XBOOLE_0:def 10;
	hence thesis by Th7;
	end;
	end;

	theorem
	for R being compatible Equivalence_Relation of free_magma(X) holds
	(free_magma X)./.R =
	the_submagma_generated_by (nat_hom R).: (canon_image(X,1) .: X)
	proof
	let R be compatible Equivalence_Relation of free_magma(X);
	set A = canon_image(X,1) .: X;
	reconsider A as Subset of free_magma X;
	A1: the carrier of the_submagma_generated_by A
	= the carrier of free_magma X by Th36;
	the carrier of (free_magma X)./.R = rng nat_hom R by FUNCT_2:def 3; then
	the carrier of (free_magma X)./.R = (nat_hom R) .: dom(nat_hom R)
	by RELAT_1:113; then
	the carrier of (free_magma X)./.R =
	(nat_hom R).: the carrier of the_submagma_generated_by A
	by A1,FUNCT_2:def 1; then
	the carrier of (free_magma X)./.R =
	the carrier of the_submagma_generated_by (nat_hom R).: A by Th15;
	hence thesis by Th7;
	end;

	theorem Th38:
	for f being Function of X,Y
	holds canon_image(Y,1)*f is Function of X, free_magma Y
	proof
	let f be Function of X,Y;
	A1: dom f = X by FUNCT_2:def 1;
	dom canon_image(Y,1) = Y by Lm3; then
	rng f c= dom canon_image(Y,1); then
	A2: dom(canon_image(Y,1)*f) = X by A1,RELAT_1:27;
	rng(canon_image(Y,1)*f) c= rng canon_image(Y,1) by RELAT_1:26;
	hence thesis by A2,FUNCT_2:2;
	end;

	definition
	let X be non empty set;
	let M be non empty multMagma;
	let n,m be non zero Nat;
	let f be Function of free_magma(X,n),M;
	let g be Function of free_magma(X,m),M;
	func [:f,g:] ->
	Function of [:[:free_magma(X,n),free_magma(X,m):],{n}:], M means :Def20:
	for x being Element of [:[:free_magma(X,n),free_magma(X,m):],{n}:],
	y being Element of free_magma(X,n), z being Element of free_magma(X,m)
	st y = x`1`1 & z = x`1`2 holds it.x = f.y * g.z;
	existence
	proof
	set X1 = [:[:free_magma(X,n),free_magma(X,m):],{n}:];
	defpred P[object,object] means
	for x being Element of X1,
	y being Element of free_magma(X,n), z being Element of free_magma(X,m)
	st $1=x & y = x`1`1 & z = x`1`2 holds $2 = f.y * g.z;
	A1: for x being object st x in X1
	ex y being object st y in the carrier of M & P[x,y]
	proof
	let x be object;
	assume x in X1; then
	A2: x`1 in [:free_magma(X,n),free_magma(X,m):] by MCART_1:10; then
	reconsider x1 = x`1`1 as Element of free_magma(X,n) by MCART_1:10;
	reconsider x2 = x`1`2 as Element of free_magma(X,m) by A2,MCART_1:10;
	set y = f.x1 * g.x2;
	take y;
	thus y in the carrier of M;
	thus P[x,y];
	end;
	consider h be Function of X1, the carrier of M such that
	A3: for x being object st x in X1 holds P[x,h.x] from FUNCT_2:sch 1(A1);
	take h;
	thus thesis by A3;
	end;
	uniqueness
	proof
	let f1,f2 be Function of [:[:free_magma(X,n),free_magma(X,m):],{n}:], M;
	assume A4:
	for x being Element of [:[:free_magma(X,n),free_magma(X,m):],{n}:],
	y being Element of free_magma(X,n), z being Element of free_magma(X,m) st
	y = x`1`1 & z = x`1`2 holds f1.x = f.y * g.z;
	assume A5:
	for x being Element of [:[:free_magma(X,n),free_magma(X,m):],{n}:],
	y being Element of free_magma(X,n), z being Element of free_magma(X,m) st
	y = x`1`1 & z = x`1`2 holds f2.x = f.y * g.z;
	for x being object st x in [:[:free_magma(X,n),free_magma(X,m):],{n}:]
	holds f1.x = f2.x
	proof
	let x be object;
	assume x in [:[:free_magma(X,n),free_magma(X,m):],{n}:]; then
	reconsider x9=x as
	Element of [:[:free_magma(X,n),free_magma(X,m):],{n}:];
	A6: x9`1 in [:free_magma(X,n),free_magma(X,m):] by MCART_1:10; then
	reconsider x1 = x9`1`1 as Element of free_magma(X,n) by MCART_1:10;
	reconsider x2 = x9`1`2 as Element of free_magma(X,m) by A6,MCART_1:10;
	thus f1.x = f.x1 * g.x2 by A4 .= f2.x by A5;
	end;
	hence thesis by FUNCT_2:12;
	end;
	end;

	reserve M for non empty multMagma;

	:: Ch I ?7.1 Pro.1 Algebra I Bourbaki

	theorem Th39:
	for f being Function of X,M holds
	ex h being Function of free_magma X, M st h is multiplicative &
	h extends f*(canon_image(X,1)")
	proof
	let f be Function of X,M;
	defpred P1[object,object] means ex n st n=$1 &
	$2 = Funcs(free_magma(X,n),the carrier of M);
	A1: for x being object st x in NAT ex y being object st P1[x,y]
	proof
	let x be object;
	assume x in NAT; then
	reconsider n=x as Nat;
	set y = Funcs(free_magma(X,n),the carrier of M);
	take y;
	thus P1[x,y];
	end;
	consider F1 be Function such that
	A2: dom F1 = NAT &
	for x being object st x in NAT holds P1[x,F1.x] from CLASSES1:sch 1(A1);
	A3: f in Funcs(X,the carrier of M) by FUNCT_2:8;
	P1[1,F1.1] by A2; then
	F1.1 = Funcs(X,the carrier of M) by Def13; then
	Funcs(X,the carrier of M) in rng F1 by A2,FUNCT_1:3; then
	A4: f in union rng F1 by A3,TARSKI:def 4; then
	A5: f in Union F1 by CARD_3:def 4;
	reconsider X1 = Union F1 as non empty set by A4,CARD_3:def 4;
	defpred P2[object,object] means
	for fs being XFinSequence of X1
	st $1=fs holds
	(((for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M) implies (
	(dom fs = 0 implies $2 = {}) &
	(dom fs = 1 implies $2 = f) &
	for n being Nat st n>=2 & dom fs = n holds
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& fs1.p = [: f1, f2 :]) & $2 = Union fs1)) &
	(not (for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M) implies $2 = f));
	A6: for e being object st e in X1^omega ex u being object st P2[e,u]
	proof
	let e be object;
	assume e in X1^omega; then
	reconsider fs = e as XFinSequence of X1 by AFINSQ_1:def 7;
	per cases;
	suppose A7: for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M;
	dom fs = 0 or dom fs + 1 > 0+1 by XREAL_1:6; then
	dom fs = 0 or dom fs >= 1 by NAT_1:13; then
	dom fs = 0 or dom fs = 1 or dom fs > 1 by XXREAL_0:1; then
	A8: dom fs = 0 or dom fs = 1 or dom fs + 1 > 1+1 by XREAL_1:6;
	per cases by A8,NAT_1:13;
	suppose A9: dom fs = 0; set u = {};
	take u; thus P2[e,u] by A9; end;
	suppose A10: dom fs = 1; set u = f;
	take u; thus P2[e,u] by A10; end;
	suppose A11: dom fs >= 2;
	reconsider n = dom fs as Nat;
	reconsider n9= n -' 1 as Nat;
	n-1 >= 2-1 by A11,XREAL_1:9; then
	A12: n9 = n-1 by XREAL_0:def 2;
	A13: Seg n9 c= Segm(n9+1) by AFINSQ_1:3;
	defpred P3[set,object] means
	for p being Nat st p>=1 & p<=n-1 & $1=p holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& $2 = [: f1, f2 :];
	A14: for k being Nat st k in Seg n9 ex x being object st P3[k,x]
	proof
	let k be Nat;
	assume A15: k in Seg n9; then
	A16: 1<=k & k<=n9 by FINSEQ_1:1; then
	k+1<=n-1+1 by A12,XREAL_1:7; then
	A17: k+1-k<=n-k by XREAL_1:9;
	then A18: n-'k = n-k by XREAL_0:def 2;
	reconsider m1=k as non zero Nat by A15,FINSEQ_1:1;
	reconsider m2=n-k as non zero Nat by A17,A18;
	reconsider f1=fs.m1 as Function of free_magma(X,m1),M
	by A7,A15,A13,A12;
	-1>=-k by A16,XREAL_1:24; then
	-1+n>=-k+n by XREAL_1:7; then
	m2 in Seg n9 by A12,A17,FINSEQ_1:1; then
	reconsider f2=fs.m2 as Function of free_magma(X,m2),M by A7,A13,A12;
	set x = [: f1, f2 :];
	take x;
	thus thesis;
	end;
	consider fs1 be FinSequence such that
	A19: dom fs1 = Seg n9 &
	for k being Nat st k in Seg n9 holds P3[k,fs1.k]
	from FINSEQ_1:sch 1(A14);
	set u = Union fs1;
	take u;
	now
	assume for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M;
	thus (dom fs = 0 implies u = {}) & (dom fs = 1 implies u = f) by A11;
	thus for n being Nat st n>=2 & dom fs = n holds
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& fs1.p = [: f1, f2 :]) & u = Union fs1
	proof
	let n99 be Nat;
	assume n99>=2;
	assume A20: dom fs = n99;
	take fs1;
	thus len fs1 = n99-1 by A12,A20,A19,FINSEQ_1:def 3;
	thus for p being Nat st p>=1 & p<=n99-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n99-p & f1=fs.m1 & f2=fs.m2 & fs1.p = [: f1, f2 :]

	by A20,FINSEQ_1:1,A12,A19;
	thus u = Union fs1;
	end;
	end;
	hence thesis by A7;
	end;
	end;
	suppose A21: not for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M;
	take f; thus thesis by A21;
	end;
	end;
	consider F2 be Function such that
	A22: dom F2 = X1^omega &
	for e being object st e in X1^omega holds P2[e,F2.e]
	from CLASSES1:sch 1(A6);
	A23: for n being Nat, fs being XFinSequence of X1
	st n>=2 & dom fs = n & (for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M) &
	(ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2 & fs1.p = [: f1, f2 :]) &
	F2.fs = Union fs1) holds
	F2.fs in Funcs(free_magma(X,n),the carrier of M)
	proof
	let n be Nat;
	let fs be XFinSequence of X1;
	assume A24: n>=2;
	assume dom fs = n;
	assume for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M;
	assume ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2 & fs1.p = [: f1, f2 :]) &
	F2.fs = Union fs1; then
	consider fs1 be FinSequence such that
	A25: len fs1 = n-1 and
	A26: for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2 & fs1.p = [: f1, f2 :] and
	A27: F2.fs = Union fs1;
	A28: for x being object st x in F2.fs ex y,z being object st x = [y,z]
	proof
	let x be object;
	assume x in F2.fs; then
	x in union rng fs1 by A27,CARD_3:def 4; then
	consider Y be set such that
	A29: x in Y & Y in rng fs1 by TARSKI:def 4;
	consider p be object such that
	A30: p in dom fs1 & Y = fs1.p by A29,FUNCT_1:def 3;
	reconsider p as Nat by A30;
	p in Seg len fs1 by A30,FINSEQ_1:def 3; then
	1<=p & p<=n-1 by A25,FINSEQ_1:1; then
	consider m1,m2 be non zero Nat,
	f1 be Function of free_magma(X,m1),M,
	f2 be Function of free_magma(X,m2),M such that
	A31: m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2 & fs1.p = [: f1, f2 :] by A26;
	consider y,z be object such that
	A32: x = [y,z] by A29,A30,A31,RELAT_1:def 1;
	take y,z;
	thus x = [y,z] by A32;
	end;
	for x,y1,y2 being object st [x,y1] in F2.fs & [x,y2] in F2.fs holds y1=y2
	proof
	let x,y1,y2 be object;
	assume [x,y1] in F2.fs; then
	[x,y1] in union rng fs1 by A27,CARD_3:def 4; then
	consider Y1 be set such that
	A33: [x,y1] in Y1 & Y1 in rng fs1 by TARSKI:def 4;
	consider p1 be object such that
	A34: p1 in dom fs1 & Y1 = fs1.p1 by A33,FUNCT_1:def 3;
	reconsider p1 as Nat by A34;
	p1 in Seg len fs1 by A34,FINSEQ_1:def 3; then
	1<=p1 & p1<=n-1 by A25,FINSEQ_1:1; then
	consider m19,m29 be non zero Nat,
	f19 be Function of free_magma(X,m19),M,
	f29 be Function of free_magma(X,m29),M such that
	A35: m19=p1 & m29 = n-p1 & f19=fs.m19 & f29=fs.m29
	& fs1.p1 = [: f19, f29 :] by A26;
	A36: x in dom [: f19, f29 :] by A33,A34,A35,FUNCT_1:1;
	then x`2 in {m19} by MCART_1:10; then
	A37: x`2 = m19 by TARSKI:def 1;
	assume [x,y2] in F2.fs; then
	[x,y2] in union rng fs1 by A27,CARD_3:def 4; then
	consider Y2 be set such that
	A38: [x,y2] in Y2 & Y2 in rng fs1 by TARSKI:def 4;
	consider p2 be object such that
	A39: p2 in dom fs1 & Y2 = fs1.p2 by A38,FUNCT_1:def 3;
	reconsider p2 as Nat by A39;
	p2 in Seg len fs1 by A39,FINSEQ_1:def 3; then
	1<=p2 & p2<=n-1 by A25,FINSEQ_1:1; then
	consider m199,m299 be non zero Nat,
	f199 be Function of free_magma(X,m199),M,
	f299 be Function of free_magma(X,m299),M such that
	A40: m199=p2 & m299 = n-p2 & f199=fs.m199 & f299=fs.m299
	& fs1.p2 = [: f199, f299 :] by A26;
	A41: x in dom [: f199, f299 :] by A38,A39,A40,FUNCT_1:1;
	then x`2 in {m199} by MCART_1:10;
	then A42: f19 = f199 & f29 = f299 by A35,A40,A37,TARSKI:def 1;
	A43: x`1 in [:free_magma(X,m19),free_magma(X,m29):] by A36,MCART_1:10;
	reconsider x1=x as
	Element of [:[:free_magma(X,m19),free_magma(X,m29):],{m19}:] by A36;
	reconsider y19=x`1`1 as Element of free_magma(X,m19) by A43,MCART_1:10;
	reconsider z1=x`1`2 as Element of free_magma(X,m29) by A43,MCART_1:10;
	A44: x`1 in [:free_magma(X,m199),free_magma(X,m299):] by A41,MCART_1:10;
	reconsider x2=x as
	Element of [:[:free_magma(X,m199),free_magma(X,m299):],{m199}:] by A41;
	reconsider y29=x`1`1 as Element of free_magma(X,m199) by A44,MCART_1:10;
	reconsider z2=x`1`2 as Element of free_magma(X,m299) by A44,MCART_1:10;
	thus y1 = [: f19, f29 :].x1 by A33,A34,A35,FUNCT_1:1
	.= f19.y19 * f29.z1 by Def20
	.= f199.y29 * f299.z2 by A42
	.= [: f199, f299 :].x2 by Def20
	.= y2 by A38,A39,A40,FUNCT_1:1;
	end; then
	reconsider f9=F2.fs as Function by A28,FUNCT_1:def 1,RELAT_1:def 1;
	for x being object holds x in free_magma(X,n) iff
	ex y being object st [x,y] in f9
	proof
	let x be object;
	hereby
	assume x in free_magma(X,n); then
	consider p,m be Nat such that
	A45: x`2 = p & 1<=p & p<=n-1 &
	x`1`1 in free_magma(X,p) & x`1`2 in free_magma(X,m) &
	n = p + m &
	x in [:[:free_magma(X,p),free_magma(X,m):],{p}:] by A24,Th21;
	consider m1,m2 be non zero Nat,
	f1 be Function of free_magma(X,m1),M,
	f2 be Function of free_magma(X,m2),M such that
	A46: m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& fs1.p = [: f1, f2 :] by A26,A45;
	reconsider x9 = x as Element of
	[:[:free_magma(X,m1),free_magma(X,m2):],{m1}:] by A45,A46;
	reconsider y9 = x`1`1 as Element of free_magma(X,m1) by A45,A46;
	reconsider z9 = x`1`2 as Element of free_magma(X,m2) by A45,A46;
	reconsider y = f1.y9 * f2.z9 as object;
	A47: dom [: f1, f2 :] = [:[:free_magma(X,m1),free_magma(X,m2):],{m1}:]
	by FUNCT_2:def 1;
	A48: [: f1, f2 :].x9 = y by Def20;
	take y;
	A49: [x,y] in fs1.p by A46,A47,A48,FUNCT_1:1;
	p in Seg len fs1 by A45,A25,FINSEQ_1:1; then
	p in dom fs1 by FINSEQ_1:def 3; then
	fs1.p in rng fs1 by FUNCT_1:3; then
	[x,y] in union rng fs1 by A49,TARSKI:def 4;
	hence [x,y] in f9 by A27,CARD_3:def 4;
	end;
	given y being object such that
	A50: [x,y] in f9;
	[x,y] in union rng fs1 by A27,A50,CARD_3:def 4; then
	consider Y be set such that
	A51: [x,y] in Y & Y in rng fs1 by TARSKI:def 4;
	consider p be object such that
	A52: p in dom fs1 & Y = fs1.p by A51,FUNCT_1:def 3;
	A53: p in Seg len fs1 by A52,FINSEQ_1:def 3;
	reconsider p as Nat by A52;
	p >= 1 & p <= n-1 by A53,A25,FINSEQ_1:1; then
	consider m1,m2 be non zero Nat,
	f1 be Function of free_magma(X,m1),M,
	f2 be Function of free_magma(X,m2),M such that
	A54: m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2 & fs1.p = [: f1, f2 :] by A26;
	A55: x in dom [: f1, f2 :] by A51,A52,A54,FUNCT_1:1;
	then A56: x`1 in [:free_magma(X,m1),free_magma(X,m2):] & x`2 in {m1}
	by MCART_1:10; then
	A57: x`1`1 in free_magma(X,m1) & x`1`2 in free_magma(X,m2) by MCART_1:10;
	x = [x`1,x`2] by A55,MCART_1:21; then
	A58: x = [[x`1`1,x`1`2],x`2] by A56,MCART_1:21;
	x`2 = m1 by A56,TARSKI:def 1; then
	x in free_magma(X,m1+m2) by A58,Th22,A57;
	hence x in free_magma(X,n) by A54;
	end; then
	A59: dom f9 = free_magma(X,n) by XTUPLE_0:def 12;
	for y being object st y in rng f9 holds y in the carrier of M
	proof
	let y be object;
	assume y in rng f9; then
	consider x being object such that
	A60: x in dom f9 & y = f9.x by FUNCT_1:def 3;
	[x,y] in Union fs1 by A27,A60,FUNCT_1:1; then
	[x,y] in union rng fs1 by CARD_3:def 4; then
	consider Y be set such that
	A61: [x,y] in Y & Y in rng fs1 by TARSKI:def 4;
	consider p be object such that
	A62: p in dom fs1 & Y = fs1.p by A61,FUNCT_1:def 3;
	A63: p in Seg len fs1 by A62,FINSEQ_1:def 3;
	reconsider p as Nat by A62;
	p >= 1 & p <= n-1 by A63,A25,FINSEQ_1:1; then
	consider m1,m2 be non zero Nat,
	f1 be Function of free_magma(X,m1),M,
	f2 be Function of free_magma(X,m2),M such that
	A64: m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2 & fs1.p = [: f1, f2 :] by A26;
	y in rng [: f1, f2 :] by A61,A62,A64,XTUPLE_0:def 13;
	hence y in the carrier of M;
	end; then
	rng f9 c= the carrier of M;
	hence thesis by A59,FUNCT_2:def 2;
	end;
	for e being object st e in X1^omega holds F2.e in X1
	proof
	let e be object;
	assume A65: e in X1^omega;
	then reconsider fs=e as XFinSequence of X1 by AFINSQ_1:def 7;
	per cases;
	suppose A66: for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M; then
	A67: (dom fs = 0 implies F2.e = {}) & (dom fs = 1 implies F2.e = f) &
	for n being Nat st n>=2 & dom fs = n holds
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& fs1.p = [: f1, f2 :]) & F2.e = Union fs1 by A65,A22;
	dom fs = 0 or dom fs + 1 > 0+1 by XREAL_1:6; then
	dom fs = 0 or dom fs >= 1 by NAT_1:13; then
	dom fs = 0 or dom fs = 1 or dom fs > 1 by XXREAL_0:1; then
	A68: dom fs = 0 or dom fs = 1 or dom fs + 1 > 1+1 by XREAL_1:6;
	per cases by A68,NAT_1:13;
	suppose A69: dom fs = 0;
	Funcs({},the carrier of M) = {{}} by FUNCT_5:57; then
	A70: {} in Funcs({},the carrier of M) by TARSKI:def 1;
	P1[0,F1.0] by A2; then
	F1.0 = Funcs({},the carrier of M) by Def13; then
	Funcs({},the carrier of M) in rng F1 by A2,FUNCT_1:3; then
	{} in union rng F1 by A70,TARSKI:def 4;
	hence F2.e in X1 by A69,A67,CARD_3:def 4;
	end;
	suppose dom fs = 1;
	hence F2.e in X1 by A5,A66,A65,A22;
	end;
	suppose A71: dom fs >= 2;
	set n=dom fs;
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& fs1.p = [: f1, f2 :]) & F2.e = Union fs1 by A66,A71,A65,A22; then
	A72: F2.e in Funcs(free_magma(X,n),the carrier of M) by A23,A71,A66;
	A73: n in dom F1 by A2,ORDINAL1:def 12; then
	P1[n,F1.n] by A2; then
	Funcs(free_magma(X,n),the carrier of M) in rng F1
	by A73,FUNCT_1:3; then
	F2.e in union rng F1 by A72,TARSKI:def 4;
	hence F2.e in X1 by CARD_3:def 4;
	end;
	end;
	suppose not (for m being non zero Nat st m in dom fs
	holds fs.m is Function of free_magma(X,m),M);
	hence F2.e in X1 by A5,A65,A22;
	end;
	end; then
	reconsider F2 as Function of X1^omega, X1 by A22,FUNCT_2:3;
	deffunc FX(XFinSequence of X1) = F2.$1;
	consider F3 be sequence of X1 such that
	A74: for n being Nat holds F3.n = FX(F3\|n)
	from FuncRecursiveExist2;
	A75: for n being Nat st n>0
	holds F3.n is Function of free_magma(X,n),M
	proof
	defpred P4[Nat] means for n being Nat st $1 = n & n > 0
	holds F3.n is Function of free_magma(X,n),M;
	A76: for k being Nat st for n being Nat st n < k holds P4[n] holds P4[k]
	proof
	let k be Nat;
	assume A77: for n being Nat st n < k holds P4[n];
	thus P4[k]
	proof
	let n be Nat;
	assume A78: k = n;
	assume n > 0;
	A79: for m being non zero Nat st m in dom(F3\|n)
	holds (F3\|n).m is Function of free_magma(X,m),M
	proof
	let m be non zero Nat;
	assume A80: m in dom(F3\|n);
	then A81: (F3\|n).m = F3.m by FUNCT_1:47;
	m in Segm k by A78,A80; then
	m < k by NAT_1:44;
	hence (F3\|n).m is Function of free_magma(X,m),M by A81,A77;
	end;
	A82: F3\|n in X1^omega by AFINSQ_1:def 7;
	reconsider fs=F3\|n as XFinSequence of X1;
	A83: (dom fs = 0 implies F2.fs = {}) & (dom fs = 1 implies F2.fs = f) &
	for n being Nat st n>=2 & dom fs = n holds
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& fs1.p = [: f1, f2 :]) & F2.fs = Union fs1 by A79,A82,A22;
	A84: n in NAT by ORDINAL1:def 12;
	dom F3 = NAT by FUNCT_2:def 1; then
	A85: n c= dom F3 by A84,ORDINAL1:def 2;
	A86: dom fs = dom F3 /\ n by RELAT_1:61
	.= n by A85,XBOOLE_1:28;
	F2.fs is Function of free_magma(X,n),M
	proof
	n = 0 or n + 1 > 0+1 by XREAL_1:6; then
	n = 0 or n >= 1 by NAT_1:13; then
	n = 0 or n = 1 or n > 1 by XXREAL_0:1; then
	A87: n = 0 or n = 1 or n + 1 > 1+1 by XREAL_1:6;
	per cases by A87,NAT_1:13;
	suppose A88: n = 0;
	Funcs({},the carrier of M) = {{}} by FUNCT_5:57; then
	F2.fs in Funcs({},the carrier of M) by A88,A83,TARSKI:def 1; then
	F2.fs in Funcs(free_magma(X,n),the carrier of M) by A88,Def13; then
	ex f being Function st F2.fs = f & dom f = free_magma(X,n)
	& rng f c= the carrier of M by FUNCT_2:def 2;
	hence thesis by FUNCT_2:2;
	end;
	suppose A89: n = 1;
	free_magma(X,1) = X by Def13;
	hence thesis by A79,A82,A22,A89,A86;
	end;
	suppose A90: n >= 2; then
	ex fs1 being FinSequence st len fs1 = n-1 &
	(for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=fs.m1 & f2=fs.m2
	& fs1.p = [: f1, f2 :]) & F2.fs = Union fs1
	by A79,A82,A22,A86; then
	F2.fs in Funcs(free_magma(X,n),the carrier of M)
	by A23,A90,A86,A79; then
	ex f being Function st F2.fs = f & dom f = free_magma(X,n)
	& rng f c= the carrier of M by FUNCT_2:def 2;
	hence thesis by FUNCT_2:2;
	end;
	end;
	hence F3.n is Function of free_magma(X,n),M by A74;
	end;
	end;
	for k being Nat holds P4[k] from NAT_1:sch 4(A76);
	hence thesis;
	end;
	reconsider X9 = the carrier of free_magma X as set;
	defpred P5[object,object] means
	for w being Element of free_magma X,
	f9 being Function of free_magma(X,w`2),M st w = $1 & f9 = F3.w`2
	holds $2 = f9.w`1;
	A91: for x being object st x in X9
	ex y being object st y in the carrier of M & P5[x,y]
	proof
	let x be object;
	assume x in X9; then
	reconsider w=x as Element of free_magma X;
	reconsider f9=F3.w`2 as Function of free_magma(X,w`2),M by A75;
	set y = f9.w`1;
	take y;
	w in [:free_magma(X,w`2),{w`2}:] by Th25; then
	w`1 in free_magma(X,w`2) by MCART_1:10;
	hence y in the carrier of M by FUNCT_2:5;
	thus P5[x,y];
	end;
	consider h be Function of X9,the carrier of M such that
	A92: for x being object st x in X9 holds P5[x,h.x] from FUNCT_2:sch 1(A91);
	reconsider h as Function of free_magma X, M;
	take h;
	for a,b being Element of free_magma X holds h.(a * b) = h.a * h.b
	proof
	let a,b be Element of free_magma X;
	reconsider fab=F3.(ab)`2 as Function of free_magma(X,(ab)`2),M by A75;
	a*b = [[[a`1,b`1],a`2],length a + length b] by Th31; then
	A93: (ab)`1=[[a`1,b`1],a`2] & (ab)`2 = length a + length b; then
	A94: fab = F2.(F3\|(length a + length b)) by A74;
	A95: F3\|(length a + length b) in X1^omega by AFINSQ_1:def 7;
	A96: for m being non zero Nat
	st m in dom(F3\|(length a + length b))
	holds (F3\|(length a + length b)).m is Function of free_magma(X,m),M
	proof
	let m be non zero Nat;
	assume A97: m in dom(F3\|(length a + length b));
	F3.m is Function of free_magma(X,m),M by A75;
	hence thesis by A97,FUNCT_1:47;
	end;
	set n = length a + length b;
	length a >= 1 & length b >= 1 by Th32; then
	A98: length a + length b >= 1+1 by XREAL_1:7;
	A99: n in NAT by ORDINAL1:def 12;
	dom F3 = NAT by FUNCT_2:def 1; then
	A100: n c= dom F3 by A99,ORDINAL1:def 2;
	dom(F3\|n) = dom F3 /\ n by RELAT_1:61
	.= n by A100,XBOOLE_1:28; then
	consider fs1 be FinSequence such that
	A101: len fs1 = n-1 and
	A102: for p being Nat st p>=1 & p<=n-1 holds
	ex m1,m2 being non zero Nat,
	f1 being Function of free_magma(X,m1),M,
	f2 being Function of free_magma(X,m2),M
	st m1=p & m2 = n-p & f1=(F3\|n).m1 & f2=(F3\|n).m2
	& fs1.p = [: f1, f2 :] and
	A103: fab = Union fs1 by A96,A98,A95,A22,A94;
	ab in [:free_magma(X,(ab)`2),{(a*b)`2}:] by Th25; then
	(ab)`1 in free_magma(X,(ab)`2) by MCART_1:10; then
	(a*b)`1 in dom fab by FUNCT_2:def 1; then
	[(ab)`1,fab.(ab)`1] in Union fs1 by A103,FUNCT_1:1; then
	[(ab)`1,fab.(ab)`1] in union rng fs1 by CARD_3:def 4; then
	consider Y be set such that
	A104: [(ab)`1,fab.(ab)`1] in Y & Y in rng fs1 by TARSKI:def 4;
	consider p be object such that
	A105: p in dom fs1 & Y = fs1.p by A104,FUNCT_1:def 3;
	A106: p in Seg len fs1 by A105,FINSEQ_1:def 3;
	reconsider p as Nat by A105;
	p >= 1 & p <= n-1 by A106,A101,FINSEQ_1:1; then
	consider m1,m2 be non zero Nat,
	f1 be Function of free_magma(X,m1),M,
	f2 be Function of free_magma(X,m2),M such that
	A107: m1=p & m2 = n-p & f1=(F3\|n).m1 & f2=(F3\|n).m2
	& fs1.p = [: f1, f2 :] by A102;
	A108: (a*b)`1 in dom [: f1, f2 :] by A107,A104,A105,FUNCT_1:1;
	then (ab)`1`1 in [:free_magma(X,m1),free_magma(X,m2):] & (ab)`1`2 in {m1}
	by MCART_1:10; then
	A109: [a`1,b`1] in [:free_magma(X,m1),free_magma(X,m2):] & a`2 in {m1}
	by A93; then
	m1 = a`2 by TARSKI:def 1; then
	A110: m1 = length a by Def18;
	length b >= 0+1 by Th32; then
	length b + length a > 0 + length a by XREAL_1:6; then
	A111: m1 in Segm n by A110,NAT_1:44;
	length a >= 0+1 by Th32; then
	length a + length b > 0 + length b by XREAL_1:6; then
	A112: m2 in Segm n by A110,A107,NAT_1:44;
	reconsider x=(a*b)`1
	as Element of [:[:free_magma(X,m1),free_magma(X,m2):],{m1}:] by A108;
	A113: x`1 in [:free_magma(X,m1),free_magma(X,m2):] by MCART_1:10; then
	reconsider y = x`1`1 as Element of free_magma(X,m1) by MCART_1:10;
	reconsider z = x`1`2 as Element of free_magma(X,m2) by A113,MCART_1:10;
	A114: x`1 = [a`1,b`1] by A93;
	A115: [: f1, f2 :].x = f1.y * f2.z by Def20;
	A116: h.(ab) = fab.(ab)`1 by A92;
	A117: fab.(a*b)`1 = [: f1, f2 :].x by A107,A104,A105,FUNCT_1:1;
	reconsider fa=F3.a`2 as Function of free_magma(X,a`2),M by A75;
	reconsider fb=F3.b`2 as Function of free_magma(X,b`2),M by A75;
	f1 = F3.m1 by A107,A111,FUNCT_1:49
	.= fa by A109,TARSKI:def 1; then
	A118: fa.a`1 = f1.y by A114;
	f2 = F3.m2 by A107,A112,FUNCT_1:49
	.= fb by Def18,A110,A107; then
	A119: fb.b`1 = f2.z by A114;
	h.b = fb.b`1 by A92;
	hence h.(a * b) = h.a * h.b by A115,A116,A118,A119,A92,A117;
	end;
	hence h is multiplicative by GROUP_6:def 6;
	set fX = canon_image(X,1);
	for x being object st x in dom(f*(fX")) holds x in dom h
	proof
	let x be object;
	assume A120: x in dom(f*(fX"));
	dom(f*(fX")) c= dom(fX") by RELAT_1:25; then
	x in dom(fX") by A120; then
	x in rng fX by FUNCT_1:33; then
	x in the carrier of free_magma X;
	hence x in dom h by FUNCT_2:def 1;
	end; then
	A121: dom(f*(fX")) c= dom h;
	for x being object st x in (dom h) /\ dom(f(fX")) holds h.x = (f(fX")).x
	proof
	let x be object;
	assume x in (dom h) /\ dom(f*(fX")); then
	A122: x in dom(f*(fX")) by A121,XBOOLE_1:28;
	A123: dom(f*(fX")) c= dom(fX") by RELAT_1:25; then
	x in dom(fX") by A122;
	then x in rng fX by FUNCT_1:33;
	then consider x9 be object such that
	A124: x9 in dom fX & x = fX.x9 by FUNCT_1:def 3;
	A125: 1 in {1} by TARSKI:def 1;
	[:free_magma(X,1),{1}:] c= free_magma_carrier X by Lm1; then
	A126: [:X,{1}:] c= free_magma_carrier X by Def13;
	A127: x9 in X by A124,Lm3;
	A128: x = [x9,1] by A124,Def19; then
	x in [:X,{1}:] by A125,A127,ZFMISC_1:def 2; then
	reconsider w=x as Element of free_magma X by A126;
	A129: (fX").x = x9 by A124,FUNCT_1:34;
	set f9 = F3.w`2;
	reconsider f9 as Function of free_magma(X,w`2),M by A75;
	A130: f9 = F3.1 by A128 .= FX(F3\|1) by A74;
	A131: for m being non zero Nat st m in dom(F3\|1)
	holds (F3\|1).m is Function of free_magma(X,m),M
	proof
	let m be non zero Nat;
	assume m in dom(F3\|1);
	then (F3\|1).m = F3.m by FUNCT_1:47;
	hence (F3\|1).m is Function of free_magma(X,m),M by A75;
	end;
	A132: F3\|1 in X1^omega by AFINSQ_1:def 7;
	reconsider fs=F3\|1 as XFinSequence of X1;
	dom F3 = NAT by FUNCT_2:def 1; then
	A133: 1 c= dom F3 by ORDINAL1:def 2;
	A134: dom fs = dom F3 /\ 1 by RELAT_1:61
	.= 1 by A133,XBOOLE_1:28;
	thus h.x = f9.w`1 by A92 .= f9.x9 by A128
	.= f.((fX").x) by A129,A130,A134,A131,A132,A22
	.= (f*(fX")).x by A123,A122,FUNCT_1:13;
	end; then
	h tolerates f*(fX") by PARTFUN1:def 4;
	hence h extends f*(canon_image(X,1)") by A121;
	end;

	theorem Th40:
	for f being Function of X,M, h,g being Function of free_magma X, M
	st h is multiplicative & h extends f*(canon_image(X,1)") &
	g is multiplicative & g extends f*(canon_image(X,1)") holds h = g
	proof
	let f be Function of X,M;
	let h,g be Function of free_magma X, M;
	assume A1: h is multiplicative;
	assume A2: h extends f*(canon_image(X,1)");
	assume A3: g is multiplicative;
	assume A4: g extends f*(canon_image(X,1)");
	defpred P[Nat] means for w being Element of free_magma X st length w = $1
	holds h.w=g.w;
	A5: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
	proof
	let k be Nat;
	assume A6: for n being Nat st n < k holds P[n];
	thus for w being Element of free_magma X st length w = k holds h.w=g.w
	proof
	let w be Element of free_magma X;
	assume A7: length w = k;
	A8: w = [w`1,w`2] & length w >= 1 by Th32; then
	length w = 1 or length w > 1 by XXREAL_0:1; then
	A9: length w = 1 or length w +1 > 1+1 by XREAL_1:8;
	per cases by A9,NAT_1:13;
	suppose A10: length w = 1;
	set x = w`1;
	x in {w9`1 where w9 is Element of free_magma X: length w9 = 1}
	by A10; then
	A11: x in X by Th30;
	A12: dom(f*(canon_image(X,1)")) c= dom h &
	h tolerates f*(canon_image(X,1)") by A2;
	A13: dom(f*(canon_image(X,1)")) c= dom g &
	g tolerates f*(canon_image(X,1)") by A4;
	A14: canon_image(X,1).x = [x,1] by A11,Lm3 .= w by Def18,A8,A10;
	x in dom canon_image(X,1) by A11,Lm3; then
	w in rng canon_image(X,1) by A14,FUNCT_1:3; then
	A15: w in dom(canon_image(X,1)") by FUNCT_1:33;
	X c= dom f by FUNCT_2:def 1; then
	dom canon_image(X,1) c= dom f by Lm3; then
	rng(canon_image(X,1)") c= dom f by FUNCT_1:33; then
	w in dom(f*(canon_image(X,1)")) by A15,RELAT_1:27; then
	w in dom h /\ dom(f*(canon_image(X,1)")) &
	w in dom g /\ dom(f*(canon_image(X,1)")) by A12,A13,XBOOLE_1:28; then
	h.w = (f(canon_image(X,1)")).w & g.w = (f(canon_image(X,1)")).w
	by A12,A13,PARTFUN1:def 4;
	hence thesis;
	end;
	suppose length w >= 2; then
	consider w1,w2 be Element of free_magma X such that
	A16: w = w1*w2 & length w1 < length w & length w2 < length w by Th34;
	h.w1 = g.w1 & h.w2 = g.w2 by A6,A7,A16; then
	h.(w1w2) = g.w1g.w2 by A1,GROUP_6:def 6;
	hence h.w=g.w by A16,A3,GROUP_6:def 6;
	end;
	end;
	end;
	A17: for k being Nat holds P[k] from NAT_1:sch 4(A5);
	for w being Element of free_magma X holds h.w=g.w
	proof
	let w be Element of free_magma X;
	reconsider k=length w as Nat;
	P[k] by A17;
	hence h.w=g.w;
	end;
	then for x being object st x in the carrier of free_magma X holds h.x = g.x;
	hence h = g by FUNCT_2:12;
	end;

	reserve M,N for non empty multMagma,
	f for Function of M, N,
	H for non empty multSubmagma of N,
	R for compatible Equivalence_Relation of M;

	theorem Th41:
	f is multiplicative & the carrier of H = rng f & R = equ_kernel f implies
	ex g being Function of M./.R, H st f = g * nat_hom R &
	g is bijective & g is multiplicative
	proof
	assume A1: f is multiplicative;
	assume A2: the carrier of H = rng f;
	assume A3: R = equ_kernel f;
	set g = ((nat_hom R)~) * f;
	for x,y1,y2 being object st [x,y1] in g & [x,y2] in g holds y1 = y2
	proof
	let x,y1,y2 be object;
	assume [x,y1] in g; then
	consider z1 be object such that
	A4: [x,z1] in (nat_hom R)~ & [z1,y1] in f by RELAT_1:def 8;
	assume [x,y2] in g; then
	consider z2 be object such that
	A5: [x,z2] in (nat_hom R)~ & [z2,y2] in f by RELAT_1:def 8;
	A6: [z1,x] in nat_hom R & [z2,x] in nat_hom R by A4,A5,RELAT_1:def 7; then
	z1 in dom nat_hom R & z2 in dom nat_hom R by XTUPLE_0:def 12;
	then reconsider z1,z2 as Element of M;
	A7: x = (nat_hom R).z1 & x = (nat_hom R).z2 by A6,FUNCT_1:1;
	A8: f.z1 = y1 & f.z2 = y2 by A4,A5,FUNCT_1:1;
	(nat_hom R).z1 = Class(R,z1) & (nat_hom R).z2 = Class(R,z2) by Def6; then
	z2 in Class(R,z1) by A7,EQREL_1:23; then
	[z1,z2] in equ_kernel f by A3,EQREL_1:18;
	hence y1 = y2 by A8,Def8;
	end; then
	reconsider g as Function by FUNCT_1:def 1;
	rng nat_hom R = the carrier of M./.R by FUNCT_2:def 3; then
	A9: dom((nat_hom R)~) = the carrier of M./.R by RELAT_1:20;
	the carrier of M c= dom f by FUNCT_2:def 1; then
	dom nat_hom R c= dom f; then
	rng((nat_hom R)~) c= dom f by RELAT_1:20; then
	A10: dom g = the carrier of M./.R by A9,RELAT_1:27;
	dom f c= the carrier of M; then
	dom f c= dom(nat_hom R) by FUNCT_2:def 1; then
	dom f c= rng((nat_hom R)~) by RELAT_1:20; then
	A11: rng g = the carrier of H by A2,RELAT_1:28;
	then reconsider g as Function of M./.R, H by A10,FUNCT_2:1;
	take g;
	for x1,x2 being object st x1 in dom g & x2 in dom g & g.x1 = g.x2
	holds x1 = x2
	proof
	let x1,x2 be object;
	assume A12: x1 in dom g;
	assume A13: x2 in dom g;
	assume A14: g.x1 = g.x2;
	set y=g.x1;
	[x1,y] in g by A12,FUNCT_1:1; then
	consider z1 be object such that
	A15: [x1,z1] in (nat_hom R)~ & [z1,y] in f by RELAT_1:def 8;
	[x2,y] in g by A14,A13,FUNCT_1:1; then
	consider z2 be object such that
	A16: [x2,z2] in (nat_hom R)~ & [z2,y] in f by RELAT_1:def 8;
	A17: [z1,x1] in nat_hom R & [z2,x2] in nat_hom R
	by A15,A16,RELAT_1:def 7; then
	z1 in dom nat_hom R & z2 in dom nat_hom R by XTUPLE_0:def 12;
	then reconsider z1,z2 as Element of M;
	z1 in dom f & z2 in dom f & f.z1=y & f.z2=y by A15,A16,FUNCT_1:1; then
	[z1,z2] in equ_kernel f by Def8; then
	A18: z2 in Class(R,z1) by A3,EQREL_1:18;
	A19: (nat_hom R).z1 = Class(R,z1) & (nat_hom R).z2 = Class(R,z2) by Def6;
	x1 = (nat_hom R).z1 & x2 = (nat_hom R).z2 by A17,FUNCT_1:1;
	hence x1 = x2 by A19,A18,EQREL_1:23;
	end; then
	A20: g is one-to-one by FUNCT_1:def 4;
	A21: for x being object
	holds x in dom f iff x in dom nat_hom R & (nat_hom R).x in dom g
	proof
	let x be object;
	hereby
	assume x in dom f; then
	x in the carrier of M;
	hence x in dom nat_hom R by FUNCT_2:def 1; then
	(nat_hom R).x in rng nat_hom R by FUNCT_1:3; then
	(nat_hom R).x in the carrier of M./.R;
	hence (nat_hom R).x in dom g by FUNCT_2:def 1;
	end;
	assume x in dom nat_hom R & (nat_hom R).x in dom g; then
	x in the carrier of M;
	hence x in dom f by FUNCT_2:def 1;
	end;
	for x being object st x in dom f holds f.x = g.((nat_hom R).x)
	proof
	let x be object;
	assume A22: x in dom f;
	set y = (nat_hom R).x;
	y in dom g by A22,A21; then
	[y,g.y] in g by FUNCT_1:1; then
	consider z be object such that
	A23: [y,z] in (nat_hom R)~ & [z,g.y] in f by RELAT_1:def 8;
	[z,y] in nat_hom R by A23,RELAT_1:def 7; then
	A24: z in dom nat_hom R & y = (nat_hom R).z by FUNCT_1:1;
	A25: z in dom f & g.y = f.z by A23,FUNCT_1:1; then
	reconsider z9=z,x9=x as Element of M by A22;
	(nat_hom R).z9 = Class(R,z9) & (nat_hom R).x9 = Class(R,x9) by Def6; then
	z9 in Class (R,x9) by A24,EQREL_1:23; then
	[x,z] in R by EQREL_1:18;
	hence f.x = g.((nat_hom R).x) by A25,A3,Def8;
	end;
	hence f = g * nat_hom R by A21,FUNCT_1:10;
	g is onto by A11,FUNCT_2:def 3;
	hence g is bijective by A20;
	for v,w being Element of M./.R holds g.(vw) = g.v g.w
	proof
	let v,w be Element of M./.R;
	v*w in the carrier of M./.R; then
	v*w in dom g by FUNCT_2:def 1; then
	[vw,g.(vw)] in g by FUNCT_1:1; then
	consider z be object such that
	A26: [vw,z] in (nat_hom R)~ & [z,g.(vw)] in f by RELAT_1:def 8;
	[z,v*w] in nat_hom R by A26,RELAT_1:def 7; then
	A27: z in dom nat_hom R & (nat_hom R).z = v*w by FUNCT_1:1;
	A28: f.z = g.(v*w) by A26,FUNCT_1:1;
	v in the carrier of M./.R; then
	v in dom g by FUNCT_2:def 1; then
	[v,g.v] in g by FUNCT_1:1; then
	consider z1 be object such that
	A29: [v,z1] in (nat_hom R)~ & [z1,g.v] in f by RELAT_1:def 8;
	[z1,v] in nat_hom R by A29,RELAT_1:def 7; then
	A30: z1 in dom nat_hom R & (nat_hom R).z1 = v by FUNCT_1:1;
	A31: f.z1 = g.v by A29,FUNCT_1:1;
	w in the carrier of M./.R; then
	w in dom g by FUNCT_2:def 1; then
	[w,g.w] in g by FUNCT_1:1; then
	consider z2 be object such that
	A32: [w,z2] in (nat_hom R)~ & [z2,g.w] in f by RELAT_1:def 8;
	[z2,w] in nat_hom R by A32,RELAT_1:def 7; then
	A33: z2 in dom nat_hom R & (nat_hom R).z2 = w by FUNCT_1:1;
	A34: f.z2 = g.w by A32,FUNCT_1:1;
	reconsider z1,z2,z as Element of M by A30,A33,A27;
	A35: (nat_hom R).z = (nat_hom R).(z1*z2) by A30,A33,A27,GROUP_6:def 6;
	(nat_hom R).(z1z2) = Class(R,z1z2) & (nat_hom R).z = Class(R,z)
	by Def6; then
	z1*z2 in Class(R,z) by A35,EQREL_1:23; then
	[z,z1*z2] in R by EQREL_1:18; then
	A36: f.z = f.(z1*z2) by A3,Def8
	.= f.z1 * f.z2 by A1,GROUP_6:def 6;
	A37: [g.v,g.w] in [:the carrier of H,the carrier of H:] by ZFMISC_1:def 2;
	thus g.(v*w) = (the multF of N).[g.v,g.w] by A31,A34,A36,A28,BINOP_1:def 1
	.= ((the multF of N)\|[:the carrier of H,the carrier of H:]).[g.v,g.w]
	by A37,FUNCT_1:49
	.= ((the multF of N)\|[:the carrier of H,the carrier of H:]).(g.v,g.w)
	by BINOP_1:def 1
	.= ((the multF of N)\|\|the carrier of H).(g.v,g.w) by REALSET1:def 2
	.= g.v * g.w by Def9;
	end;
	hence g is multiplicative by GROUP_6:def 6;
	end;

	theorem
	for g1,g2 being Function of M./.R, N st g1 * nat_hom R = g2 * nat_hom R
	holds g1 = g2
	proof
	let g1,g2 be Function of M./.R, N;
	assume A1: g1 * nat_hom R = g2 * nat_hom R;
	set Y = rng nat_hom R;
	rng nat_hom R = the carrier of M ./. R by FUNCT_2:def 3; then
	dom g1 = Y & dom g2 = Y by FUNCT_2:def 1;
	hence g1 = g2 by A1,FUNCT_1:86;
	end;

	theorem
	for M being non empty multMagma holds ex X being non empty set,
	r being Relators of free_magma X,
	g being Function of (free_magma X) ./. equ_rel r, M
	st g is bijective & g is multiplicative
	proof
	let M be non empty multMagma;
	set X = the carrier of M;
	consider f be Function of free_magma X, M such that
	A1: f is multiplicative & f extends (id X)*(canon_image(X,1)") by Th39;
	consider r be Relators of free_magma X such that
	A2: equ_kernel f = equ_rel r by A1,Th5;
	reconsider R = equ_kernel f
	as compatible Equivalence_Relation of free_magma X by A1,Th4;
	the multF of M = (the multF of M)\|[:the carrier of M,the carrier of M:]; then
	the multF of M = (the multF of M)\|\|the carrier of M by REALSET1:def 2; then
	reconsider H = M as non empty multSubmagma of M by Def9;
	for y being object st y in the carrier of M
	ex x being object st x in dom f & y = f.x
	proof
	let y be object;
	assume A3: y in the carrier of M;
	reconsider x = [y,1] as set;
	take x;
	[:free_magma(X,1),{1}:] c= the carrier of free_magma X by Lm1; then
	A4: [:X,{1}:] c= the carrier of free_magma X by Def13;
	1 in {1} by TARSKI:def 1; then
	x in [:X,{1}:] by A3,ZFMISC_1:def 2; then
	x in the carrier of free_magma X by A4;
	hence x in dom f by FUNCT_2:def 1;
	A5: dom ((id X)*(canon_image(X,1)")) c= dom f &
	f tolerates (id X)*(canon_image(X,1)") by A1;
	A6: canon_image(X,1).y = x by A3,Lm3;
	y in dom canon_image(X,1) by A3,Lm3; then
	x in rng canon_image(X,1) by A6,FUNCT_1:3; then
	A7: x in dom(canon_image(X,1)") by FUNCT_1:33;
	dom canon_image(X,1) c= dom(id X) by Lm3; then
	rng(canon_image(X,1)") c= dom(id X) by FUNCT_1:33; then
	A8: x in dom((id X)*(canon_image(X,1)")) by A7,RELAT_1:27;
	A9: y in dom canon_image(X,1) by A3,Lm3;
	thus y = (id X).y by A3,FUNCT_1:18
	.= (id X).((canon_image(X,1)").x) by A9,A6,FUNCT_1:34
	.= ((id X)*(canon_image(X,1)")).x by A8,FUNCT_1:12
	.= f.x by A8,A5,PARTFUN1:53;
	end; then
	the carrier of M c= rng f by FUNCT_1:9; then
	the carrier of H = rng f by XBOOLE_0:def 10; then
	consider g be Function of (free_magma X) ./. R, H such that
	A10: f = g * nat_hom R & g is bijective & g is multiplicative by A1,Th41;
	reconsider g as Function of (free_magma X) ./. equ_rel r, M by A2;
	take X,r,g;
	thus thesis by A10,A2;
	end;

	definition
	let X,Y be non empty set;
	let f be Function of X,Y;
	func free_magmaF f -> Function of free_magma X, free_magma Y means :Def21:
	it is multiplicative & it extends (canon_image(Y,1)f)(canon_image(X,1)");
	existence
	proof
	reconsider f9=canon_image(Y,1)*f as Function of X, free_magma Y by Th38;
	ex h being Function of free_magma X, free_magma Y st h is multiplicative &
	h extends f9*(canon_image(X,1)") by Th39;
	hence thesis;
	end;
	uniqueness
	proof
	let f1, f2 be Function of free_magma X,free_magma Y;
	assume A1: f1 is multiplicative &
	f1 extends (canon_image(Y,1)f)(canon_image(X,1)");
	assume A2: f2 is multiplicative &
	f2 extends (canon_image(Y,1)f)(canon_image(X,1)");
	reconsider f9=canon_image(Y,1)*f as Function of X,free_magma Y by Th38;
	f1 extends f9*(canon_image(X,1)") &
	f2 extends f9*(canon_image(X,1)") by A1,A2;
	hence f1 = f2 by A1,A2,Th40;
	end;
	end;

	registration
	let X,Y be non empty set;
	let f be Function of X,Y;
	cluster free_magmaF f -> multiplicative;
	coherence by Def21;
	end;

	reserve f for Function of X,Y;
	reserve g for Function of Y,Z;

	theorem Th44:
	free_magmaF(gf) = free_magmaF(g)free_magmaF(f)
	proof
	set f2=free_magmaF(g)*free_magmaF(f);
	reconsider f9=canon_image(Z,1)(gf) as Function of X,free_magma Z by Th38;
	for a, b being Element of free_magma X holds f2.(ab) = f2.a f2.b
	proof
	let a, b be Element of free_magma X;
	a*b in the carrier of free_magma X; then
	A1: a*b in dom f2 by FUNCT_2:def 1;
	a in the carrier of free_magma X & b in the carrier of free_magma X; then
	A2: a in dom(free_magmaF f) & b in dom(free_magmaF f) by FUNCT_2:def 1;
	thus f2.(ab) = (free_magmaF g).((free_magmaF f).(ab)) by A1,FUNCT_1:12
	.= (free_magmaF g).((free_magmaF f).a * (free_magmaF f).b)
	by GROUP_6:def 6
	.= (free_magmaF g).((free_magmaF f).a)*(free_magmaF g).((free_magmaF f).b)
	by GROUP_6:def 6
	.= f2.a * (free_magmaF g).((free_magmaF f).b) by A2,FUNCT_1:13
	.= f2.a * f2.b by A2,FUNCT_1:13;
	end; then
	A3: f2 is multiplicative by GROUP_6:def 6;
	A4: dom(f9*(canon_image(X,1)")) c= dom(canon_image(X,1)") by RELAT_1:25;
	rng canon_image(X,1) c= the carrier of free_magma X; then
	dom(canon_image(X,1)") c= the carrier of free_magma X by FUNCT_1:33; then
	dom(f9*(canon_image(X,1)")) c= the carrier of free_magma X
	by A4; then
	A5: dom(f9*(canon_image(X,1)")) c= dom f2 by FUNCT_2:def 1;
	for x being object st x in dom(f9*(canon_image(X,1)"))
	holds f2.x = (f9*(canon_image(X,1)")).x
	proof
	let x be object;
	assume A6: x in dom(f9*(canon_image(X,1)"));
	free_magmaF(f) extends (canon_image(Y,1)f)(canon_image(X,1)")
	by Def21; then
	A7: dom((canon_image(Y,1)f)(canon_image(X,1)")) c= dom(free_magmaF f) &
	(canon_image(Y,1)f)(canon_image(X,1)") tolerates free_magmaF f;
	A8: x in dom(canon_image(X,1)") by A6,FUNCT_1:11;
	X c= dom f by FUNCT_2:def 1; then
	dom canon_image(X,1) c= dom f by Lm3; then
	rng(canon_image(X,1)") c= dom f by FUNCT_1:33; then
	A9: x in dom(f*(canon_image(X,1)")) by A8,RELAT_1:27;
	rng(f*(canon_image(X,1)")) c= Y; then
	rng(f*(canon_image(X,1)")) c= dom canon_image(Y,1) by Lm3; then
	x in dom(canon_image(Y,1)(f(canon_image(X,1)"))) by A9,RELAT_1:27; then
	A10: x in dom((canon_image(Y,1)f)(canon_image(X,1)")) by RELAT_1:36;
	set y = (f*(canon_image(X,1)")).x;
	free_magmaF(g) extends (canon_image(Z,1)g)(canon_image(Y,1)")
	by Def21; then
	A11: dom((canon_image(Z,1)g)(canon_image(Y,1)")) c= dom(free_magmaF g) &
	(canon_image(Z,1)g)(canon_image(Y,1)") tolerates free_magmaF g;
	y in rng(f*(canon_image(X,1)")) by A9,FUNCT_1:3; then
	A12: y in Y; then
	A13: y in dom canon_image(Y,1) by Lm3; then
	A14: canon_image(Y,1).y in rng canon_image(Y,1) by FUNCT_1:3;
	Y c= dom g by FUNCT_2:def 1; then
	A15: dom canon_image(Y,1) c= dom g by Lm3;
	rng g c= Z; then
	rng g c= dom canon_image(Z,1) by Lm3; then
	dom canon_image(Y,1) c= dom(canon_image(Z,1)*g) by A15,RELAT_1:27; then
	A16: rng(canon_image(Y,1)") c= dom(canon_image(Z,1)*g) by FUNCT_1:33;
	rng canon_image(Y,1) c= dom(canon_image(Y,1)") by FUNCT_1:33; then
	A17: rng canon_image(Y,1) c=
	dom((canon_image(Z,1)g)(canon_image(Y,1)")) by A16,RELAT_1:27;
	A18: rng canon_image(Y,1) c= dom(canon_image(Y,1)") by FUNCT_1:33;
	dom canon_image(Y,1) = Y by Lm3; then
	A19: y in dom((canon_image(Y,1)")*canon_image(Y,1)) by A12,A18,RELAT_1:27;
	A20: (canon_image(Z,1)g)(f*(canon_image(X,1)"))
	= canon_image(Z,1)(g(f*(canon_image(X,1)"))) by RELAT_1:36
	.= canon_image(Z,1)((gf)*(canon_image(X,1)")) by RELAT_1:36
	.= (canon_image(Z,1)(gf))*(canon_image(X,1)") by RELAT_1:36;
	thus f2.x = (free_magmaF g).((free_magmaF f).x) by A6,A5,FUNCT_1:12
	.= (free_magmaF g).(((canon_image(Y,1)f)(canon_image(X,1)")).x)
	by A10,A7,PARTFUN1:53
	.= (free_magmaF g).((canon_image(Y,1)(f(canon_image(X,1)"))).x)
	by RELAT_1:36
	.= (free_magmaF g).(canon_image(Y,1).((f*(canon_image(X,1)")).x))
	by A9,FUNCT_1:13
	.= ((canon_image(Z,1)g)(canon_image(Y,1)")).(canon_image(Y,1).y)
	by A17,A14,A11,PARTFUN1:53
	.= (((canon_image(Z,1)g)(canon_image(Y,1)"))*canon_image(Y,1)).y
	by A13,FUNCT_1:13
	.= ((canon_image(Z,1)g)((canon_image(Y,1)")*canon_image(Y,1))).y
	by RELAT_1:36
	.= (canon_image(Z,1)g).(((canon_image(Y,1)")canon_image(Y,1)).y)
	by A19,FUNCT_1:13
	.= (canon_image(Z,1)g).((f(canon_image(X,1)")).x) by A13,FUNCT_1:34
	.= (f9*(canon_image(X,1)")).x by A20,A9,FUNCT_1:13;
	end; then
	f2 tolerates f9*(canon_image(X,1)") by A5,PARTFUN1:53; then
	f2 extends f9*(canon_image(X,1)") by A5;
	hence free_magmaF(gf) = free_magmaF(g)free_magmaF(f) by Def21,A3;
	end;

	theorem Th45:
	for g being Function of X,Z st Y c= Z & f=g
	holds free_magmaF f = free_magmaF g
	proof
	let g be Function of X,Z;
	assume A1: Y c= Z;
	assume A2: f = g;
	A3: the carrier of free_magma Y c= the carrier of free_magma Z
	by A1,Th27; then
	reconsider f9=free_magmaF f as Function of free_magma X, free_magma Z
	by FUNCT_2:7;
	for a, b being Element of free_magma X holds f9.(a * b) = (f9.a) * (f9.b)
	proof
	let a, b be Element of free_magma X;
	set v = (free_magmaF f).a;
	set w = (free_magmaF f).b;
	reconsider v9=v, w9=w as Element of free_magma Z by A3;
	A4: length v = v`2 by Def18 .= length v9 by Def18;
	A5: length w = w`2 by Def18 .= length w9 by Def18;
	thus f9.(a * b) = (free_magmaF f).a * (free_magmaF f).b by GROUP_6:def 6
	.= [[[v9`1,w9`1],v9`2],(length v9) + (length w9)] by Th31,A4,A5
	.= (f9.a) * (f9.b) by Th31;
	end; then
	A6: f9 is multiplicative by GROUP_6:def 6;
	rng g c= Z; then
	rng g c= dom canon_image(Z,1) by Lm3; then
	A7: dom (canon_image(Z,1)*g) = dom g by RELAT_1:27;
	X c= dom g by FUNCT_2:def 1; then
	dom canon_image(X,1) c= dom (canon_image(Z,1)*g) by Lm3,A7; then
	rng (canon_image(X,1)") c= dom (canon_image(Z,1)*g) by FUNCT_1:33; then
	A8: dom((canon_image(Z,1)g)(canon_image(X,1)")) =
	dom (canon_image(X,1)") by RELAT_1:27;
	rng canon_image(X,1) c= the carrier of free_magma X; then
	dom((canon_image(Z,1)g)(canon_image(X,1)")) c=
	the carrier of free_magma X by A8,FUNCT_1:33; then
	A9: dom((canon_image(Z,1)g)(canon_image(X,1)")) c= dom f9
	by FUNCT_2:def 1;
	for x being object st x in dom((canon_image(Z,1)g)(canon_image(X,1)"))
	holds f9.x = ((canon_image(Z,1)g)(canon_image(X,1)")).x
	proof
	let x be object;
	assume A10: x in dom((canon_image(Z,1)g)(canon_image(X,1)"));
	free_magmaF(f) extends (canon_image(Y,1)f)(canon_image(X,1)")
	by Def21; then
	A11: dom((canon_image(Y,1)f)(canon_image(X,1)")) c= dom(free_magmaF f) &
	(canon_image(Y,1)f)(canon_image(X,1)") tolerates free_magmaF f;
	rng f c= Y; then
	A12: rng f c= dom canon_image(Y,1) by Lm3;
	rng f c= Z by A1; then
	A13: rng f c= dom canon_image(Z,1) by Lm3;
	A14: dom(canon_image(Y,1)*f) = dom f by A12,RELAT_1:27
	.= dom(canon_image(Z,1)*f) by A13,RELAT_1:27;
	for x being object st x in dom(canon_image(Y,1)*f)
	holds (canon_image(Y,1)f).x = (canon_image(Z,1)f).x
	proof
	let x be object;
	assume A15: x in dom(canon_image(Y,1)*f); then
	A16: f.x in dom canon_image(Y,1) by FUNCT_1:11; then
	A17: f.x in Y by Lm3;
	thus (canon_image(Y,1)*f).x = canon_image(Y,1).(f.x) by A15,FUNCT_1:12
	.= [f.x,1] by A16,Def19
	.= canon_image(Z,1).(f.x) by A1,A17,Lm3
	.= (canon_image(Z,1)*f).x by A14,A15,FUNCT_1:12;
	end; then
	canon_image(Y,1)f = canon_image(Z,1)g by A2,A14,FUNCT_1:2;
	hence f9.x = ((canon_image(Z,1)g)(canon_image(X,1)")).x
	by A10,A11,PARTFUN1:53;
	end; then
	f9 tolerates (canon_image(Z,1)g)(canon_image(X,1)")
	by A9,PARTFUN1:53; then
	f9 extends (canon_image(Z,1)g)(canon_image(X,1)") by A9;
	hence free_magmaF f = free_magmaF g by A6,Def21;
	end;

	theorem Th46:
	free_magmaF id X = id dom free_magmaF f
	proof
	dom free_magmaF id X = the carrier of free_magma X by FUNCT_2:def 1; then
	A1: dom free_magmaF id X = dom free_magmaF f by FUNCT_2:def 1;
	for x being object st x in dom free_magmaF f holds (free_magmaF id X).x = x
	proof
	let x be object;
	assume A2: x in dom free_magmaF f;
	defpred P[Nat] means for w being Element of free_magma X st length w = $1
	holds (free_magmaF id X).w = w;
	A3: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
	proof
	let k be Nat;
	assume A4: for n being Nat st n < k holds P[n];
	thus for w being Element of free_magma X st length w = k
	holds (free_magmaF id X).w = w
	proof
	let w be Element of free_magma X;
	assume A5: length w = k;
	A6: w = [w`1,w`2] & length w >= 1 by Th32; then
	length w = 1 or length w > 1 by XXREAL_0:1; then
	A7: length w = 1 or length w +1 > 1+1 by XREAL_1:8;
	per cases by A7,NAT_1:13;
	suppose A8: length w = 1;
	set y = w`1;
	y in {w9`1 where w9 is Element of free_magma X: length w9 = 1}
	by A8; then
	A9: y in X by Th30; then
	A10: y in dom id X;
	(free_magmaF id X) extends
	(canon_image(X,1)id X)(canon_image(X,1)") by Def21; then
	A11: dom((canon_image(X,1)id X)(canon_image(X,1)")) c=
	dom(free_magmaF id X) &
	(canon_image(X,1)id X)(canon_image(X,1)") tolerates
	free_magmaF id X;
	A12: canon_image(X,1).y = [y,1] by A9,Lm3 .= w by Def18,A6,A8;
	A13: y in dom canon_image(X,1) by A9,Lm3; then
	w in rng canon_image(X,1) by A12,FUNCT_1:3; then
	A14: w in dom(canon_image(X,1)") by FUNCT_1:33;
	rng id X = X.= dom canon_image(X,1) by Lm3; then
	dom(canon_image(X,1)*id X) = dom id X by RELAT_1:27; then
	X = dom(canon_image(X,1)*id X); then
	dom canon_image(X,1) c= dom(canon_image(X,1)*id X) by Lm3; then
	rng(canon_image(X,1)") c= dom(canon_image(X,1)*id X)
	by FUNCT_1:33; then
	A15: w in dom((canon_image(X,1)id X)(canon_image(X,1)"))
	by A14,RELAT_1:27;
	(canon_image(X,1)").w = y by A13,A12,FUNCT_1:34;
	then ((canon_image(X,1)id X)(canon_image(X,1)")).w
	= (canon_image(X,1)*id X).y by A15,FUNCT_1:12
	.= canon_image(X,1).((id X).y) by A10,FUNCT_1:13
	.= w by A12,A9,FUNCT_1:18;
	hence (free_magmaF id X).w = w by A15,A11,PARTFUN1:53;
	end;
	suppose length w >= 2; then
	consider w1,w2 be Element of free_magma X such that
	A16: w = w1*w2 & length w1 < length w & length w2 < length w by Th34;
	thus (free_magmaF id X).w
	= (free_magmaF id X).w1 * (free_magmaF id X).w2
	by A16,GROUP_6:def 6
	.= w1 * (free_magmaF id X).w2 by A4,A5,A16
	.= w by A4,A5,A16;
	end;
	end;
	end;
	A17: for k being Nat holds P[k] from NAT_1:sch 4(A3);
	for w being Element of free_magma X holds (free_magmaF id X).w = w
	proof
	let w be Element of free_magma X;
	reconsider k=length w as Nat;
	P[k] by A17;
	hence (free_magmaF id X).w = w;
	end;
	hence (free_magmaF id X).x = x by A2;
	end;
	hence free_magmaF id X = id dom free_magmaF f by A1,FUNCT_1:17;
	end;

	:: Ch I ?7.1 Pro.2 Algebra I Bourbaki

	theorem
	f is one-to-one implies free_magmaF f is one-to-one
	proof
	assume A1: f is one-to-one; then
	A2: f"*f = id dom f by FUNCT_1:39;
	set Y9 = rng f;
	dom f = X by FUNCT_2:def 1; then
	reconsider f1=f as Function of X, Y9 by FUNCT_2:1;
	reconsider f2=f1" as Function of Y9, X by A1,FUNCT_2:25;
	f2*f1 = id X by A2,FUNCT_2:def 1; then
	(free_magmaF f2)*(free_magmaF f1) = free_magmaF(id X) by Th44; then
	(free_magmaF f2)*(free_magmaF f) = free_magmaF(id X) by Th45; then
	(free_magmaF f2)*(free_magmaF f) = id dom(free_magmaF f) by Th46;
	hence free_magmaF f is one-to-one by FUNCT_1:31;
	end;

	:: Ch I ?7.1 Pro.2 Algebra I Bourbaki

	theorem
	f is onto implies free_magmaF f is onto
	proof
	assume A1: f is onto;
	for y being object
	st y in the carrier of free_magma Y holds y in rng free_magmaF f
	proof
	let y be object;
	assume A2: y in the carrier of free_magma Y;
	defpred P[Nat] means for w being Element of free_magma Y st length w = $1
	holds ex v being Element of free_magma X st w = (free_magmaF f).v;
	A3: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
	proof
	let k be Nat;
	assume A4: for n being Nat st n < k holds P[n];
	thus for w being Element of free_magma Y st length w = k
	holds ex v being Element of free_magma X st w = (free_magmaF f).v
	proof
	let w be Element of free_magma Y;
	assume A5: length w = k;
	A6: w = [w`1,w`2] & length w >= 1 by Th32; then
	length w = 1 or length w > 1 by XXREAL_0:1; then
	A7: length w = 1 or length w +1 > 1+1 by XREAL_1:8;
	per cases by A7,NAT_1:13;
	suppose A8: length w = 1;
	set y = w`1;
	y in {w9`1 where w9 is Element of free_magma Y: length w9 = 1}
	by A8; then
	A9: y in Y by Th30;
	(free_magmaF f) extends
	(canon_image(Y,1)f)(canon_image(X,1)") by Def21; then
	A10: dom((canon_image(Y,1)f)(canon_image(X,1)")) c=
	dom(free_magmaF f) &
	(canon_image(Y,1)f)(canon_image(X,1)") tolerates
	free_magmaF f;
	A11: canon_image(Y,1).y = [y,1] by A9,Lm3 .= w by Def18,A6,A8;
	A12: rng f = Y by A1,FUNCT_2:def 3; then
	consider x being object such that
	A13: x in dom f & y = f.x by A9,FUNCT_1:def 3;
	A14: 1 in {1} by TARSKI:def 1;
	A15: x in X by A13; then
	x in free_magma(X,1) by Def13; then
	A16: [x,1] in [:free_magma(X,1),{1}:] by A14,ZFMISC_1:def 2;
	[:free_magma(X,1),{1}:] c= free_magma_carrier X by Lm1; then
	reconsider v = [x,1] as Element of free_magma X by A16;
	take v;
	A17: x in dom canon_image(X,1) by Lm3,A15;
	A18: v = canon_image(X,1).x by Lm3,A13; then
	v in rng canon_image(X,1) by A17,FUNCT_1:3; then
	A19: v in dom(canon_image(X,1)") by FUNCT_1:33;
	rng f = dom canon_image(Y,1) by Lm3,A12; then
	dom f = dom(canon_image(Y,1)*f) by RELAT_1:27; then
	X c= dom(canon_image(Y,1)*f) by FUNCT_2:def 1; then
	dom canon_image(X,1) c= dom(canon_image(Y,1)*f) by Lm3; then
	rng(canon_image(X,1)") c= dom(canon_image(Y,1)*f) by FUNCT_1:33; then
	A20: v in dom((canon_image(Y,1)f)(canon_image(X,1)"))
	by A19,RELAT_1:27; then
	A21: (free_magmaF f).v
	= ((canon_image(Y,1)f)(canon_image(X,1)")).v by A10,PARTFUN1:53
	.= (canon_image(Y,1)*f).((canon_image(X,1)").v) by A20,FUNCT_1:12;
	x in dom canon_image(X,1) by A15,Lm3; then
	(canon_image(X,1)").v = x by A18,FUNCT_1:34;
	hence thesis by A11,A13,A21,FUNCT_1:13;
	end;
	suppose length w >= 2; then
	consider w1,w2 be Element of free_magma Y such that
	A22: w = w1*w2 & length w1 < length w & length w2 < length w by Th34;
	consider v1 be Element of free_magma X such that
	A23: w1 = (free_magmaF f).v1 by A22,A4,A5;
	consider v2 be Element of free_magma X such that
	A24: w2 = (free_magmaF f).v2 by A22,A4,A5;
	take v1*v2;
	thus thesis by A23,A24,A22,GROUP_6:def 6;
	end;
	end;
	end;
	A25: for k being Nat holds P[k] from NAT_1:sch 4(A3);
	A26: for w being Element of free_magma Y holds
	ex v being Element of free_magma X st w = (free_magmaF f).v
	proof
	let w be Element of free_magma Y;
	reconsider k=length w as Nat;
	P[k] by A25;
	hence thesis;
	end;
	ex x st x in dom free_magmaF f & y = (free_magmaF f).x
	proof
	consider x be Element of free_magma X such that
	A27: y = (free_magmaF f).x by A2,A26;
	take x;
	x in the carrier of free_magma X;
	hence x in dom free_magmaF f by FUNCT_2:def 1;
	thus y = (free_magmaF f).x by A27;
	end;
	hence y in rng free_magmaF f by FUNCT_1:def 3;
	end; then
	the carrier of free_magma Y c= rng free_magmaF f; then
	rng free_magmaF f = the carrier of free_magma Y by XBOOLE_0:def 10;
	hence free_magmaF f is onto by FUNCT_2:def 3;
	end;