:: 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 v*w in Class(R,v9*w9); 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,v*w) = Class(R,v9*w9) 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,v*w) = {} .= Class(R,v9*w9) 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 v*w in Class(R,v9*w9) by A1; hence Class(R,v*w) = Class(R,v9*w9) 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,v*w) = Class(R,v9*w9); for v,v9,w,w9 being Element of M st v in Class(R,v9) & w in Class(R,w9) holds v*w in Class(R,v9*w9) 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,v*w) = Class(R,v9*w9) by A4; hence v*w in Class(R,v9*w9) 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).(v*w) = (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).(v*w) = (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 v*w in Class(equ_rel r,v9*w9) 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 [v9*w9,v*w] 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 v*w in Class(R,v9*w9) by A6,Def3; hence [v9*w9,v*w] in Y by A5,EQREL_1:18; end; then [v9*w9,v*w] in meet X by A3,SETFAM_1:def 1; hence v*w in Class(equ_rel r,v9*w9) 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 v*w in Class(R,v9*w9) 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.(v9*w9) = 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 [v9*w9,v*w] in R by Def8; hence v*w in Class(R,v9*w9) 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 v*w 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.(v*w) = g.v*g.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: p0+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 v1*v2 = w1*w2 implies v1 = w1 & v2 = w2 proof assume A1: v1*v2 = w1*w2; 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.(a*b)`2 as Function of free_magma(X,(a*b)`2),M by A75; a*b = [[[a`1,b`1],a`2],length a + length b] by Th31; then A93: (a*b)`1=[[a`1,b`1],a`2] & (a*b)`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; a*b in [:free_magma(X,(a*b)`2),{(a*b)`2}:] by Th25; then (a*b)`1 in free_magma(X,(a*b)`2) by MCART_1:10; then (a*b)`1 in dom fab by FUNCT_2:def 1; then [(a*b)`1,fab.(a*b)`1] in Union fs1 by A103,FUNCT_1:1; then [(a*b)`1,fab.(a*b)`1] in union rng fs1 by CARD_3:def 4; then consider Y be set such that A104: [(a*b)`1,fab.(a*b)`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 (a*b)`1`1 in [:free_magma(X,m1),free_magma(X,m2):] & (a*b)`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.(a*b) = fab.(a*b)`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.(w1*w2) = g.w1*g.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.(v*w) = 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 [v*w,g.(v*w)] in g by FUNCT_1:1; then consider z be object such that A26: [v*w,z] in (nat_hom R)~ & [z,g.(v*w)] 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).(z1*z2) = Class(R,z1*z2) & (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(g*f) = free_magmaF(g)*free_magmaF(f) proof set f2=free_magmaF(g)*free_magmaF(f); reconsider f9=canon_image(Z,1)*(g*f) as Function of X,free_magma Z by Th38; for a, b being Element of free_magma X holds f2.(a*b) = 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.(a*b) = (free_magmaF g).((free_magmaF f).(a*b)) 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)*((g*f)*(canon_image(X,1)")) by RELAT_1:36 .= (canon_image(Z,1)*(g*f))*(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(g*f) = 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;