:: Homomorphisms of algebras. Quotient Universal Algebra
::  by Ma{\l}gorzata Korolkiewicz

environ

 vocabularies UNIALG_1, SUBSET_1, NUMBERS, UNIALG_2, XBOOLE_0, FINSEQ_1,
      FUNCT_1, RELAT_1, NAT_1, TARSKI, STRUCT_0, PARTFUN1, MSUALG_3, CQC_SIM1,
      WELLORD1, FINSEQ_2, GROUP_6, EQREL_1, FUNCT_2, CARD_3, RELAT_2, ALG_1;
 notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, NAT_1, RELAT_1,
      RELAT_2, FUNCT_1, RELSET_1, PARTFUN1, FINSEQ_1, EQREL_1, FINSEQ_2,
      FUNCT_2, STRUCT_0, MARGREL1, UNIALG_1, FINSEQOP, FINSEQ_3, UNIALG_2;
 constructors EQREL_1, FINSEQOP, UNIALG_2, RELSET_1, CARD_3, FINSEQ_3, CARD_1,
      NAT_1, NUMBERS;
 registrations RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, EQREL_1, FINSEQ_2,
      STRUCT_0, UNIALG_1, UNIALG_2, ORDINAL1, FINSEQ_1, CARD_1, RELSET_1,
      MARGREL1;
 requirements BOOLE, SUBSET;
 definitions UNIALG_2, RELAT_2, TARSKI, FUNCT_1, XBOOLE_0, FUNCT_2, MARGREL1;
 equalities UNIALG_2, XBOOLE_0;
 expansions UNIALG_2, FUNCT_1, FUNCT_2, MARGREL1;
 theorems FINSEQ_1, FINSEQ_2, FUNCT_1, FUNCT_2, PARTFUN1, UNIALG_1, UNIALG_2,
      RELAT_1, RELSET_1, EQREL_1, ZFMISC_1, FINSEQ_3, XBOOLE_0, RELAT_2,
      ORDERS_1, MARGREL1;
 schemes FINSEQ_1, RELSET_1, FUNCT_2, FUNCT_1;

begin

reserve U1,U2,U3 for Universal_Algebra,
  n,m for Nat,
  o1 for operation of U1,
  o2 for operation of U2,
  o3 for operation of U3,
  x,y for set;

theorem Th1:
  for B be non empty Subset of U1 st B = the carrier of U1 holds
  Opers(U1,B) = the charact of(U1)
proof
  let B be non empty Subset of U1;
A1: dom Opers(U1,B) = dom the charact of(U1) by UNIALG_2:def 6;
  assume
A2: B = the carrier of U1;
  now
    let n be Nat;
    assume
A3: n in dom the charact of(U1);
    then reconsider o = (the charact of U1).n as operation of U1 by
FUNCT_1:def 3;
    thus Opers(U1,B).n = o/.B by A1,A3,UNIALG_2:def 6
      .= (the charact of U1).n by A2,UNIALG_2:4;
  end;
  hence thesis by A1;
end;

reserve a for FinSequence of U1,
  f for Function of U1,U2;

theorem
  f*(<*>the carrier of U1) = <*>the carrier of U2;

theorem Th3:
  (id the carrier of U1)*a = a
proof
  set f = id the carrier of U1;
A1: dom (f*a) = dom a by FINSEQ_3:120;
A2: now
    let n be Nat;
    assume
A3: n in dom(f*a);
    then reconsider u = a.n as Element of U1 by A1,FINSEQ_2:11;
    f.u = u;
    hence (f*a).n = a.n by A3,FINSEQ_3:120;
  end;
  len (f*a) = len a by FINSEQ_3:120;
  hence thesis by A2,FINSEQ_2:9;
end;

theorem Th4:
  for h1 be Function of U1,U2, h2 be Function of U2,U3,a be
  FinSequence of U1 holds h2*(h1*a) = (h2 * h1)*a
proof
  let h1 be Function of U1,U2, h2 be Function of U2,U3,a be FinSequence of U1;
A1: dom a = Seg len a by FINSEQ_1:def 3;
A2: dom (h2*(h1*a)) = dom(h1*a) by FINSEQ_3:120;
  dom (h1*a) = dom a by FINSEQ_3:120;
  then
A3: dom (h2*(h1*a)) = Seg len a by A2,FINSEQ_1:def 3;
A4: len a = len((h2 * h1 qua Function of the carrier of U1, the carrier of
  U3) *(a qua FinSequence of the carrier of U1)) by FINSEQ_3:120;
  then
A5: dom ((h2 * h1)*a) = Seg len a by FINSEQ_1:def 3;
A6: now
    let n be Nat;
    assume
A7: n in dom(h2*(h1*a));
    hence (h2*(h1*a)).n = h2.((h1*a).n) by FINSEQ_3:120
      .= h2.(h1.(a.n)) by A2,A7,FINSEQ_3:120
      .= (h2*h1).(a.n) by A1,A3,A7,FINSEQ_2:11,FUNCT_2:15
      .= ((h2 * h1)*a).n by A3,A5,A7,FINSEQ_3:120;
  end;
  len(h2*(h1*a)) = len(h1*a) & len(h1*a) = len a by FINSEQ_3:120;
  hence thesis by A4,A6,FINSEQ_2:9;
end;

definition
  let U1,U2,f;
  pred f is_homomorphism means
  U1,U2 are_similar &
  for n st n in dom the charact of(U1)
  for o1,o2 st o1=(the charact of U1).n &
               o2=(the charact of U2).n
   for x be FinSequence of U1 st x in dom o1 holds f.(o1.x) = o2.(f*x);
end;

definition
  let U1,U2,f;
  pred f is_monomorphism means
  f is_homomorphism & f is one-to-one;
  pred f is_epimorphism means
  f is_homomorphism & rng f = the carrier of U2;
end;

definition
  let U1,U2,f;
  pred f is_isomorphism means
  f is_monomorphism & f is_epimorphism;
end;

definition
  let U1,U2;
  pred U1,U2 are_isomorphic means
  ex f st f is_isomorphism;
end;

theorem Th5:
  id the carrier of U1 is_homomorphism
proof
  thus U1,U1 are_similar;
  let n;
  assume n in dom the charact of(U1);
  let o1,o2 be operation of U1;
  assume
A1: o1=(the charact of U1).n & o2=(the charact of U1).n;
  set f = id the carrier of U1;
  let x be FinSequence of U1;
  assume x in dom o1;
  then o1.x in rng o1 by FUNCT_1:def 3;
  then reconsider u = o1.x as Element of U1;
  f.u = u;
  hence thesis by A1,Th3;
end;

theorem Th6:
  for h1 be Function of U1,U2, h2 be Function of U2,U3 st h1
is_homomorphism & h2 is_homomorphism holds h2 * h1 is_homomorphism
proof
  let h1 be Function of U1,U2, h2 be Function of U2,U3;
  set s1 = signature U1, s2 = signature U2, s3 = signature U3;
  assume that
A1: h1 is_homomorphism and
A2: h2 is_homomorphism;
  U1,U2 are_similar by A1;
  then
A3: s1 = s2;
  U2,U3 are_similar by A2;
  hence s1 = s3 by A3;
  let n;
  assume
A4: n in dom the charact of(U1);
  let o1,o3;
  assume that
A5: o1=(the charact of U1).n and
A6: o3=(the charact of U3).n;
  let a;
  reconsider b = h1*a as Element of (the carrier of U2)* by FINSEQ_1:def 11;
  assume
A7: a in dom o1;
  then
A8: o1.a in rng o1 by FUNCT_1:def 3;
  dom o1 = (arity o1)-tuples_on (the carrier of U1) by MARGREL1:22;
  then a in {w where w is Element of (the carrier of U1)*: len w = arity o1}
  by A7,FINSEQ_2:def 4;
  then
A9: ex w be Element of (the carrier of U1)* st w = a & len w = arity o1;
A10: len s1 = len the charact of(U1) & dom the charact of(U1) = Seg len the
  charact of(U1) by FINSEQ_1:def 3,UNIALG_1:def 4;
A11: len s2 = len the charact of(U2) & dom the charact of(U2) = Seg len the
  charact of(U2) by FINSEQ_1:def 3,UNIALG_1:def 4;
  then reconsider o2 = (the charact of U2).n as operation of U2 by A3,A10,A4,
FUNCT_1:def 3;
A12: dom s1 = Seg len s1 by FINSEQ_1:def 3;
  then
A13: s2.n = arity o2 by A3,A10,A4,UNIALG_1:def 4;
  s1.n = arity o1 by A10,A12,A4,A5,UNIALG_1:def 4;
  then len(h1*a) = arity o2 by A3,A13,A9,FINSEQ_3:120;
  then dom o2 = (arity o2)-tuples_on (the carrier of U2) & b in {s where s is
  Element of (the carrier of U2)*: len s = arity o2} by MARGREL1:22;
  then h1*a in dom o2 by FINSEQ_2:def 4;
  then
A14: h2.(o2.(h1*a)) = o3.(h2*(h1*a)) by A2,A3,A10,A11,A4,A6;
  h1.(o1.a) = o2.(h1*a) by A1,A4,A5,A7;
  hence (h2 * h1).(o1.a) = o3.(h2*(h1*a)) by A8,A14,FUNCT_2:15
    .= o3.((h2 * h1)*a) by Th4;
end;

theorem Th7:
  f is_isomorphism iff f is_homomorphism & rng f = the
  carrier of U2 & f is one-to-one
proof
  thus f is_isomorphism implies f is_homomorphism & rng f = the
  carrier of U2 & f is one-to-one
  proof
    assume f is_isomorphism;
    then f is_monomorphism & f is_epimorphism;
    hence thesis;
  end;
  assume f is_homomorphism & rng f = the carrier of U2 & f is one-to-one;
  then f is_monomorphism & f is_epimorphism;
  hence thesis;
end;

theorem Th8:
  f is_isomorphism implies dom f = the carrier of U1 & rng f
  = the carrier of U2
proof
  assume f is_isomorphism;
  then f is_epimorphism;
  hence thesis by FUNCT_2:def 1;
end;

theorem Th9:
  for h be Function of U1,U2, h1 be Function of U2,U1 st h
  is_isomorphism & h1=h" holds h1 is_homomorphism
proof
  let h be Function of U1,U2,h1 be Function of U2,U1;
  assume that
A1: h is_isomorphism and
A2: h1=h";
A3: h is one-to-one by A1,Th7;
A4: h is_homomorphism by A1,Th7;
  then
A5: U1,U2 are_similar;
  then
A6: signature U1 = signature U2;
A7: len (signature U1) = len the charact of(U1) & dom the charact of(U1) =
  Seg len the charact of(U1) by FINSEQ_1:def 3,UNIALG_1:def 4;
A8: dom (signature U2) = Seg len (signature U2) by FINSEQ_1:def 3;
A9: len (signature U2) = len the charact of(U2) & dom the charact of(U2) =
  Seg len the charact of(U2) by FINSEQ_1:def 3,UNIALG_1:def 4;
A10: rng h = the carrier of U2 by A1,Th7;
  now
    let n;
    assume
A11: n in dom the charact of(U2);
    let o2,o1;
    assume
A12: o2 = (the charact of U2).n & o1 = (the charact of U1).n;
    let x be FinSequence of U2;
    defpred P[set,set] means h.$2 = x.$1;
A13: dom x = Seg len x by FINSEQ_1:def 3;
A14: for m be Nat st m in Seg len x ex a being Element of U1 st P[m,a]
    proof
      let m be Nat;
      assume m in Seg len x;
      then m in dom x by FINSEQ_1:def 3;
      then x.m in the carrier of U2 by FINSEQ_2:11;
      then consider a be object such that
A15:  a in dom h and
A16:  h.a = x.m by A10,FUNCT_1:def 3;
      reconsider a as Element of U1 by A15;
      take a;
      thus thesis by A16;
    end;
    consider p being FinSequence of U1 such that
A17: dom p = Seg len x & for m be Nat st m in Seg len x holds P[m,p.m]
    from FINSEQ_1:sch 5(A14);
A18: dom (h*p) = dom p by FINSEQ_3:120;
    now
      let n be Nat;
      assume
A19:  n in dom x;
      hence x.n = h.(p.n) by A17,A13
        .= (h*p).n by A17,A13,A18,A19,FINSEQ_3:120;
    end;
    then
A20: x = h*p by A17,A13,A18;
A21: len p = len x by A17,FINSEQ_1:def 3;
    assume x in dom o2;
    then x in (arity o2)-tuples_on the carrier of U2 by MARGREL1:22;
    then x in {s where s is Element of (the carrier of U2)*: len s = arity o2
    } by FINSEQ_2:def 4;
    then
A22: ex s be Element of (the carrier of U2)* st x=s & len s = arity o2;
A23: (h1 * h) = (id dom h) by A2,A3,FUNCT_1:39
      .= id the carrier of U1 by FUNCT_2:def 1;
    then
A24: h1*x = (id the carrier of U1)*p by A20,Th4
      .=p by Th3;
    reconsider p as Element of (the carrier of U1)* by FINSEQ_1:def 11;
    (signature U1).n = arity o1 & (signature U2).n = arity o2 by A6,A8,A9,A11
,A12,UNIALG_1:def 4;
    then
    p in {w where w is Element of (the carrier of U1)*: len w = arity o1}
    by A6,A22,A21;
    then p in (arity o1)-tuples_on the carrier of U1 by FINSEQ_2:def 4;
    then
A25: p in dom o1 by MARGREL1:22;
    then
A26: h1.(o2.x) = h1.(h.(o1.p)) by A4,A6,A7,A9,A11,A12,A20;
A27: o1.p in the carrier of U1 by A25,PARTFUN1:4;
    then o1.p in dom h by FUNCT_2:def 1;
    hence h1.(o2.x) = (id the carrier of U1).(o1.p) by A23,A26,FUNCT_1:13
      .= o1.(h1*x) by A24,A27,FUNCT_1:17;
  end;
  hence thesis by A5;
end;

theorem Th10:
  for h be Function of U1,U2, h1 be Function of U2,U1 st h
  is_isomorphism & h1 = h" holds h1 is_isomorphism
proof
  let h be Function of U1,U2,h1 be Function of U2,U1;
  assume that
A1: h is_isomorphism and
A2: h1=h";
A3: h1 is_homomorphism by A1,A2,Th9;
A4: h is one-to-one by A1,Th7;
  then rng h1 = dom h by A2,FUNCT_1:33
    .= the carrier of U1 by FUNCT_2:def 1;
  hence thesis by A2,A4,A3,Th7;
end;

theorem Th11:
  for h be Function of U1,U2, h1 be Function of U2,U3 st h
is_isomorphism & h1 is_isomorphism holds h1 * h is_isomorphism
proof
  let h be Function of U1,U2, h1 be Function of U2,U3;
  assume that
A1: h is_isomorphism and
A2: h1 is_isomorphism;
  dom h1 = the carrier of U2 & rng h = the carrier of U2 by A1,Th8,
FUNCT_2:def 1;
  then
A3: rng (h1 * h) = rng h1 by RELAT_1:28
    .= the carrier of U3 by A2,Th8;
  h is_homomorphism & h1 is_homomorphism by A1,A2,Th7;
  then
A4: h1 * h is_homomorphism by Th6;
  h is one-to-one & h1 is one-to-one by A1,A2,Th7;
  hence thesis by A3,A4,Th7;
end;

theorem
  U1,U1 are_isomorphic
proof
  set i = id the carrier of U1;
  i is_homomorphism & rng i = the carrier of U1 by Th5;
  then i is_isomorphism by Th7;
  hence thesis;
end;

theorem
  U1,U2 are_isomorphic implies U2,U1 are_isomorphic
proof
  assume U1,U2 are_isomorphic;
  then consider f such that
A1: f is_isomorphism;
  f is_monomorphism by A1;
  then
A2: f is one-to-one;
  then
A3: rng(f") = dom f by FUNCT_1:33
    .= the carrier of U1 by FUNCT_2:def 1;
A4: f is_epimorphism by A1;
  dom(f") = rng f by A2,FUNCT_1:33
    .= the carrier of U2 by A4;
  then reconsider g = f" as Function of U2,U1 by A3,FUNCT_2:def 1,RELSET_1:4;
  take g;
  thus thesis by A1,Th10;
end;

theorem
  U1,U2 are_isomorphic & U2,U3 are_isomorphic implies U1,U3 are_isomorphic
proof
  assume U1,U2 are_isomorphic;
  then consider f such that
A1: f is_isomorphism;
  assume U2,U3 are_isomorphic;
  then consider g be Function of U2,U3 such that
A2: g is_isomorphism;
  g * f is_isomorphism by A1,A2,Th11;
  hence thesis;
end;

definition
  let U1,U2,f;
  assume
A1: f is_homomorphism;
  func Image f -> strict SubAlgebra of U2 means
  :Def6:
  the carrier of it = f .: (the carrier of U1);
  existence
  proof
A2: dom f = the carrier of U1 by FUNCT_2:def 1;
    then reconsider A = f .: (the carrier of U1) as non empty Subset of U2;
    take B = UniAlgSetClosed(A);
    A is opers_closed
    proof
      let o2 be operation of U2;
      consider n being Nat such that
A3:   n in dom the charact of(U2) and
A4:   (the charact of U2).n = o2 by FINSEQ_2:10;
      let s be FinSequence of A;
      assume
A5:   len s = arity o2;
      defpred P[object,object] means f.$2 = s.$1;
A6:   for x being object st x in dom s
       ex y being object st y in the carrier of U1 & P[x,y]
      proof
        let x be object;
        assume
A7:     x in dom s;
        then reconsider x0 = x as Element of NAT;
        s.x0 in A by A7,FINSEQ_2:11;
        then consider y being object such that
A8:     y in dom f and
        y in the carrier of U1 and
A9:     f.y = s.x0 by FUNCT_1:def 6;
        take y;
        thus thesis by A8,A9;
      end;
      consider s1 be Function such that
A10:  dom s1 = dom s & rng s1 c= the carrier of U1 &
      for x being object st x in
      dom s holds P[x,s1.x] from FUNCT_1:sch 6(A6);
      dom s1 = Seg len s by A10,FINSEQ_1:def 3;
      then reconsider s1 as FinSequence by FINSEQ_1:def 2;
      reconsider s1 as FinSequence of U1 by A10,FINSEQ_1:def 4;
      reconsider s1 as Element of (the carrier of U1)* by FINSEQ_1:def 11;
A11:  len s1 = len s by A10,FINSEQ_3:29;
A12:  dom (signature U2) = Seg len (signature U2) by FINSEQ_1:def 3;
A13:  U1,U2 are_similar by A1;
      then
A14:  signature U1 = signature U2;
A15:  dom (signature U1) = dom (signature U2) by A13;
A16:  len (signature U2) = len the charact of(U2) & dom the charact of(U2)
      = Seg len the charact of(U2) by FINSEQ_1:def 3,UNIALG_1:def 4;
      then
A17:  (signature U2).n = arity o2 by A3,A4,A12,UNIALG_1:def 4;
A18:  len (f*s1) = len s1 by FINSEQ_3:120;
A19:  dom (f*s1) = Seg len (f*s1) & dom s = Seg len s1 by A10,FINSEQ_1:def 3;
      now
        let m be Nat;
        assume
A20:    m in dom s;
        then f.(s1.m) = s.m by A10;
        hence (f*s1).m = s.m by A18,A19,A20,FINSEQ_3:120;
      end;
      then
A21:  s = f*s1 by A11,A18,FINSEQ_2:9;
A22:  dom (signature U1) = Seg len (signature U1) by FINSEQ_1:def 3;
A23:  len (signature U1) = len the charact of(U1) & dom the charact of(U1)
      = Seg len the charact of(U1) by FINSEQ_1:def 3,UNIALG_1:def 4;
      then reconsider o1 = (the charact of U1).n as operation of U1 by A3,A16
,A22,A15,A12,FUNCT_1:def 3;
      (signature U1).n = arity o1 by A3,A16,A15,A12,UNIALG_1:def 4;
      then
      s1 in {w where w is Element of (the carrier of U1)* : len w = arity
      o1 } by A14,A5,A17,A11;
      then s1 in (arity o1)-tuples_on the carrier of U1 by FINSEQ_2:def 4;
      then
A24:  s1 in dom o1 by MARGREL1:22;
      then
A25:  o1.s1 in rng o1 by FUNCT_1:def 3;
      f.(o1.s1) = o2.(f*s1) by A1,A3,A4,A23,A16,A22,A15,A12,A24;
      hence thesis by A2,A21,A25,FUNCT_1:def 6;
    end;
    then B = UAStr (# A,Opers(U2,A) #) by UNIALG_2:def 8;
    hence thesis;
  end;
  uniqueness
  proof
    let A,B be strict SubAlgebra of U2;
    reconsider A1 = the carrier of A as non empty Subset of U2
    by UNIALG_2:def 7;
    the charact of(A) = Opers(U2,A1) by UNIALG_2:def 7;
    hence thesis by UNIALG_2:def 7;
  end;
end;

theorem
  for h be Function of U1,U2 st h is_homomorphism holds rng h =
  the carrier of Image h
proof
  let h be Function of U1,U2;
  dom h = the carrier of U1 by FUNCT_2:def 1;
  then
A1: rng h = h.:(the carrier of U1) by RELAT_1:113;
  assume h is_homomorphism;
  hence thesis by A1,Def6;
end;

theorem
  for U2 being strict Universal_Algebra, f be Function of U1,U2 st f
  is_homomorphism holds f is_epimorphism iff Image f = U2
proof
  let U2 be strict Universal_Algebra;
  let f be Function of U1,U2;
  assume
A1: f is_homomorphism;
  thus f is_epimorphism implies Image f = U2
  proof
    reconsider B = the carrier of Image f as non empty Subset of U2 by
UNIALG_2:def 7;
    assume f is_epimorphism;
    then
A2: the carrier of U2 = rng f
      .= f.:(dom f) by RELAT_1:113
      .= f.:(the carrier of U1) by FUNCT_2:def 1
      .= the carrier of Image f by A1,Def6;
    the charact of(Image f) = Opers(U2,B) by UNIALG_2:def 7;
    hence thesis by A2,Th1;
  end;
  assume Image f = U2;
  then the carrier of U2 = f.:(the carrier of U1) by A1,Def6
    .= f.:(dom f) by FUNCT_2:def 1
    .= rng f by RELAT_1:113;
  hence thesis by A1;
end;

begin :: Quotient Universal Algebra

definition
  let U1 be 1-sorted;
  mode Relation of U1 is Relation of the carrier of U1;
  mode Equivalence_Relation of U1 is Equivalence_Relation of the carrier of U1;
end;

definition
  let U1;
  mode Congruence of U1 -> Equivalence_Relation of U1 means
    :Def7:
    for n,o1
    st n in dom the charact of(U1) & o1 = (the charact of U1).n for x,y be
FinSequence of U1 st x in dom o1 & y in dom o1 & [x,y] in ExtendRel(it) holds [
    o1.x,o1.y] in it;
  existence
  proof
    reconsider P = id the carrier of U1 as Equivalence_Relation of U1;
    take P;
    let n,o1;
    assume that
    n in dom the charact of(U1) and
    o1 = (the charact of U1).n;
    let x,y be FinSequence of U1;
    assume that
A1: x in dom o1 and
    y in dom o1 and
A2: [x,y] in ExtendRel(P);
    [x,y] in id ((the carrier of U1)*) by A2,FINSEQ_3:121;
    then
A3: x = y by RELAT_1:def 10;
    o1.x in rng o1 by A1,FUNCT_1:def 3;
    hence thesis by A3,RELAT_1:def 10;
  end;
end;

reserve E for Congruence of U1;

definition
  let U1 be Universal_Algebra, E be Congruence of U1, o be operation of U1;
  func QuotOp(o,E) -> homogeneous quasi_total non empty PartFunc of (Class E)*
  ,(Class E) means
  :Def8:
  dom it = (arity o)-tuples_on (Class E) & for y be
FinSequence of (Class E) st y in dom it for x be FinSequence of the carrier of
  U1 st x is_representatives_FS y holds it.y = Class(E,o.x);
  existence
  proof
    defpred P[object,object] means
   for y be FinSequence of (Class E) st y = $1 holds
for x be FinSequence of the carrier of U1 st x is_representatives_FS y holds $2
    = Class(E,o.x);
    set X = (arity o)-tuples_on (Class E);
A1: for e be object st e in X ex u be object st u in Class(E) & P[e,u]
    proof
      let e be object;
A2:   dom o = (arity o)-tuples_on the carrier of U1 by MARGREL1:22
        .={q where q is Element of (the carrier of U1)*: len q = arity o} by
FINSEQ_2:def 4;
      assume e in X;
      then e in {s where s is Element of (Class E)*: len s = arity o} by
FINSEQ_2:def 4;
      then consider s be Element of (Class E)* such that
A3:   s = e and
A4:   len s = arity o;
      consider x be FinSequence of the carrier of U1 such that
A5:   x is_representatives_FS s by FINSEQ_3:122;
      take y = Class(E,o.x);
A6:   len x = arity o by A4,A5,FINSEQ_3:def 4;
      x is Element of (the carrier of U1)* by FINSEQ_1:def 11;
      then
A7:   x in dom o by A2,A6;
      then
A8:   o.x in rng o by FUNCT_1:def 3;
      hence y in Class E by EQREL_1:def 3;
      let a be FinSequence of (Class E);
      assume
A9:   a = e;
      let b be FinSequence of the carrier of U1;
      assume
A10:  b is_representatives_FS a;
      then
A11:  len b = arity o by A3,A4,A9,FINSEQ_3:def 4;
      for m st m in dom x holds [x.m,b.m] in E
      proof
        let m;
        assume
A12:    m in dom x;
        then
A13:    Class(E,x.m) = s.m & x.m in rng x by A5,FINSEQ_3:def 4,FUNCT_1:def 3;
        dom x = Seg arity o by A6,FINSEQ_1:def 3
          .= dom b by A11,FINSEQ_1:def 3;
        then Class(E,b.m) = s.m by A3,A9,A10,A12,FINSEQ_3:def 4;
        hence thesis by A13,EQREL_1:35;
      end;
      then
A14:  [x,b] in ExtendRel(E) by A6,A11,FINSEQ_3:def 3;
      b is Element of (the carrier of U1)* by FINSEQ_1:def 11;
      then
      (ex n being Nat st n in dom the charact of(U1) & (the charact of U1
      ).n = o ) & b in dom o by A2,A11,FINSEQ_2:10;
      then [o.x,o.b] in E by A7,A14,Def7;
      hence thesis by A8,EQREL_1:35;
    end;
    consider F being Function such that
A15: dom F = X & rng F c= Class(E) & for e be object st e in X holds P[e,
    F.e] from FUNCT_1:sch 6(A1);
    X in the set of all m-tuples_on Class E;
    then X c= union the set of all m-tuples_on Class E by ZFMISC_1:74;
    then X c= (Class E)* by FINSEQ_2:108;
    then reconsider F as PartFunc of (Class E)*,Class E by A15,RELSET_1:4;
A16: dom F = {t where t is Element of (Class E)*: len t = arity o} by A15,
FINSEQ_2:def 4;
A17: for x,y be FinSequence of Class E st len x = len y & x in dom F holds
    y in dom F
    proof
      let x,y be FinSequence of Class E;
      assume that
A18:  len x = len y and
A19:  x in dom F;
A20:  y is Element of (Class E)* by FINSEQ_1:def 11;
      ex t1 be Element of (Class E)* st x = t1 & len t1 = arity o by A16,A19;
      hence thesis by A16,A18,A20;
    end;
A21:  ex x being FinSequence st x in dom F
     proof
      set a = the Element of X;
      a in X;
      hence ex x being FinSequence st x in dom F by A15;
     end;
    dom F is with_common_domain
    proof
      let x,y be FinSequence;
      assume x in dom F & y in dom F;
      then (ex t1 be Element of (Class E)* st x = t1 & len t1 = arity o )& ex
      t2 be Element of (Class E)* st y = t2 & len t2 = arity o by A16;
      hence thesis;
    end;
    then reconsider
    F as homogeneous quasi_total non empty PartFunc of (Class E)*,
    Class E by A17,A21,MARGREL1:def 21,def 22;
    take F;
    thus dom F = (arity o)-tuples_on (Class E) by A15;
    let y be FinSequence of (Class E);
    assume
A22: y in dom F;
    let x be FinSequence of the carrier of U1;
    assume x is_representatives_FS y;
    hence thesis by A15,A22;
  end;
  uniqueness
  proof
    let F,G be homogeneous quasi_total non empty PartFunc of (Class(E))*,Class
    (E);
    assume that
A23: dom F = (arity o)-tuples_on (Class E) and
A24: for y be FinSequence of Class(E) st y in dom F for x be
FinSequence of the carrier of U1 st x is_representatives_FS y holds F.y = Class
    (E,o.x) and
A25: dom G = (arity(o))-tuples_on (Class(E)) and
A26: for y be FinSequence of Class(E) st y in dom G for x be
FinSequence of the carrier of U1 st x is_representatives_FS y holds G.y = Class
    (E,o.x);
    for a be object st a in dom F holds F.a = G.a
    proof
      let a be object;
      assume
A27:  a in dom F;
      then reconsider b = a as FinSequence of Class(E) by FINSEQ_1:def 11;
      consider x be FinSequence of the carrier of U1 such that
A28:  x is_representatives_FS b by FINSEQ_3:122;
      F.b = Class(E,o.x) by A24,A27,A28;
      hence thesis by A23,A25,A26,A27,A28;
    end;
    hence thesis by A23,A25;
  end;
end;

definition
  let U1,E;
  func QuotOpSeq(U1,E) -> PFuncFinSequence of Class E means
  :Def9:
  len it =
len the charact of(U1) & for n st n in dom it for o1 st (the charact of(U1)).n
  = o1 holds it.n = QuotOp(o1,E);
  existence
  proof
    defpred P[set,set] means for o be Element of Operations(U1) st o = (the
    charact of(U1)).$1 holds $2 = QuotOp(o,E);
A1: for n be Nat st n in Seg len the charact of(U1) ex x be Element of
    PFuncs((Class E)*,(Class E)) st P[n,x]
    proof
      let n be Nat;
      assume n in Seg len the charact of(U1);
      then n in dom the charact of(U1) by FINSEQ_1:def 3;
      then reconsider o = (the charact of(U1)).n as operation of U1 by
FUNCT_1:def 3;
      reconsider x = QuotOp(o,E) as Element of PFuncs((Class E)*,(Class E)) by
PARTFUN1:45;
      take x;
      thus thesis;
    end;
    consider p be FinSequence of PFuncs((Class E)*,(Class E)) such that
A2: dom p = Seg len the charact of(U1) & for n be Nat st n in Seg len
    the charact of(U1) holds P[n,p.n] from FINSEQ_1:sch 5(A1);
    reconsider p as PFuncFinSequence of Class E;
    take p;
    thus len p = len the charact of(U1) by A2,FINSEQ_1:def 3;
    let n;
    assume n in dom p;
    hence thesis by A2;
  end;
  uniqueness
  proof
    let F,G be PFuncFinSequence of Class E;
    assume that
A3: len F = len the charact of(U1) and
A4: for n st n in dom F for o1 st (the charact of(U1)).n = o1 holds F.
    n = QuotOp(o1,E) and
A5: len G = len the charact of(U1) and
A6: for n st n in dom G for o1 st (the charact of(U1)).n = o1 holds G.
    n = QuotOp(o1,E);
    now
      let n be Nat;
      assume
A7:   n in dom F;
      dom F = Seg len the charact of(U1) by A3,FINSEQ_1:def 3;
      then n in dom the charact of(U1) by A7,FINSEQ_1:def 3;
      then reconsider o1 = (the charact of U1).n as operation of U1 by
FUNCT_1:def 3;
A8:   dom F = dom the charact of(U1) & dom G = dom the charact of(U1) by A3,A5,
FINSEQ_3:29;
      F.n = QuotOp(o1,E) by A4,A7;
      hence F.n = G.n by A6,A8,A7;
    end;
    hence thesis by A3,A5,FINSEQ_2:9;
  end;
end;

definition
  let U1,E;
  func QuotUnivAlg(U1,E) -> strict Universal_Algebra equals
  UAStr (# Class(E),QuotOpSeq(U1,E) #);
  coherence
proof
  set UU = UAStr (# Class(E),QuotOpSeq(U1,E) #);
  for n be Nat,h be PartFunc of (Class E)*,(Class E) st n in dom QuotOpSeq
  (U1,E) & h = QuotOpSeq(U1,E).n holds h is homogeneous
  proof
    let n be Nat,h be PartFunc of (Class E)*,(Class E);
    assume that
A1: n in dom QuotOpSeq(U1,E) and
A2: h = QuotOpSeq(U1,E).n;
    n in Seg len QuotOpSeq(U1,E) by A1,FINSEQ_1:def 3;
    then n in Seg len the charact of U1 by Def9;
    then n in dom the charact of U1 by FINSEQ_1:def 3;
    then reconsider o = (the charact of U1).n as operation of U1 by
FUNCT_1:def 3;
    QuotOpSeq(U1,E).n = QuotOp(o,E) by A1,Def9;
    hence thesis by A2;
  end;
  then
A3: the charact of UU is homogeneous;
  for n be Nat ,h be PartFunc of (Class E)*,(Class E) st n in dom
  QuotOpSeq(U1,E) & h = QuotOpSeq(U1,E).n holds h is quasi_total
  proof
    let n be Nat,h be PartFunc of (Class E)*,(Class E);
    assume that
A4: n in dom QuotOpSeq(U1,E) and
A5: h = QuotOpSeq(U1,E).n;
    n in Seg len QuotOpSeq(U1,E) by A4,FINSEQ_1:def 3;
    then n in Seg len the charact of(U1) by Def9;
    then n in dom the charact of U1 by FINSEQ_1:def 3;
    then reconsider o = (the charact of U1).n as operation of U1 by
FUNCT_1:def 3;
    QuotOpSeq(U1,E).n = QuotOp(o,E) by A4,Def9;
    hence thesis by A5;
  end;
  then
A6: the charact of UU is quasi_total;
  for n be object st n in dom QuotOpSeq(U1,E)
  holds QuotOpSeq(U1,E).n is non empty
  proof
    let n be object;
    assume
A7: n in dom QuotOpSeq(U1,E);
    then n in Seg len QuotOpSeq(U1,E) by FINSEQ_1:def 3;
    then n in Seg len the charact of U1 by Def9;
    then
A8: n in dom the charact of U1 by FINSEQ_1:def 3;
    reconsider n as Element of NAT by A7;
    reconsider o = (the charact of U1).n as operation of U1
    by A8,FUNCT_1:def 3;
    QuotOpSeq(U1,E).n = QuotOp(o,E) by A7,Def9;
    hence thesis;
  end;
  then
A9: the charact of UU is non-empty by FUNCT_1:def 9;
  the charact of UU <> {}
  proof
    assume
A10: the charact of UU = {};
    len the charact of UU = len the charact of U1 by Def9;
    hence contradiction by A10;
  end;
  hence thesis by A3,A6,A9,UNIALG_1:def 1,def 2,def 3;
end;
end;

definition
  let U1,E;
  func Nat_Hom(U1,E) -> Function of U1,QuotUnivAlg(U1,E) means
  :Def11:
  for u be Element of U1 holds it.u = Class(E,u);
  existence
  proof
    defpred P[Element of U1,set] means $2 = Class(E,$1);
A1: for x being Element of U1 ex y being Element of QuotUnivAlg(U1,E) st P
    [x,y]
    proof
      let x be Element of U1;
      reconsider y = Class(E,x) as Element of QuotUnivAlg(U1,E) by
EQREL_1:def 3;
      take y;
      thus thesis;
    end;
    consider f being Function of U1,QuotUnivAlg(U1,E) such that
A2: for x being Element of U1 holds P[x,f.x] from FUNCT_2:sch 3(A1);
    take f;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let f,g be Function of U1,QuotUnivAlg(U1,E);
    assume that
A3: for u be Element of U1 holds f.u = Class(E,u) and
A4: for u be Element of U1 holds g.u = Class(E,u);
    now
      let u be Element of U1;
      f.u = Class(E,u) by A3;
      hence f.u = g.u by A4;
    end;
    hence thesis;
  end;
end;

theorem Th17:
  for U1,E holds Nat_Hom(U1,E) is_homomorphism
proof
  let U1,E;
  set f = Nat_Hom(U1,E), u1 = the carrier of U1, qu = the carrier of
  QuotUnivAlg(U1,E);
A1: len (signature U1) = len the charact of(U1) by UNIALG_1:def 4;
A2: dom (signature U1) = Seg len(signature U1) by FINSEQ_1:def 3;
A3: len QuotOpSeq(U1,E) = len the charact of(U1) by Def9;
A4: len (signature QuotUnivAlg(U1,E)) = len the charact of(QuotUnivAlg(U1,E)
  ) by UNIALG_1:def 4;
  now
    let n be Nat;
    assume
A5: n in dom (signature U1);
    then n in dom the charact of(U1) by A1,A2,FINSEQ_1:def 3;
    then reconsider o1 = (the charact of U1).n as operation of U1 by
FUNCT_1:def 3;
    n in dom QuotOpSeq(U1,E) by A3,A1,A2,A5,FINSEQ_1:def 3;
    then
A6: QuotOpSeq(U1,E).n = QuotOp(o1,E) by Def9;
    reconsider cl = QuotOp(o1,E) as homogeneous quasi_total non empty PartFunc
    of qu*,qu;
    consider b be object such that
A7: b in dom cl by XBOOLE_0:def 1;
    reconsider b as Element of qu* by A7;
    dom QuotOp(o1,E) = (arity o1)-tuples_on Class(E) by Def8;
    then b in {w where w is Element of (Class(E))*: len w = arity o1} by A7,
FINSEQ_2:def 4;
    then ex w be Element of (Class(E))* st w = b & len w = arity o1;
    then
A8: arity cl = arity o1 by A7,MARGREL1:def 25;
    n in dom (signature QuotUnivAlg(U1,E)) & (signature U1).n = arity o1
    by A3,A4,A2,A5,FINSEQ_1:def 3,UNIALG_1:def 4;
    hence (signature U1).n = (signature QuotUnivAlg(U1,E)).n by A6,A8,
UNIALG_1:def 4;
  end;
  hence signature U1 = signature QuotUnivAlg(U1,E) by A3,A4,A1,FINSEQ_2:9;
  let n;
  assume n in dom the charact of(U1);
  then n in Seg len the charact of(U1) by FINSEQ_1:def 3;
  then
A9: n in dom QuotOpSeq(U1,E) by A3,FINSEQ_1:def 3;
  let o1 be operation of U1, o2 be operation of QuotUnivAlg(U1,E);
  assume
  (the charact of U1).n = o1 & o2 = (the charact of QuotUnivAlg(U1,E) ).n;
  then
A10: o2 = QuotOp(o1,E) by A9,Def9;
  let x be FinSequence of U1;
  reconsider x1 = x as Element of u1* by FINSEQ_1:def 11;
  reconsider fx = f*x as FinSequence of Class(E);
  reconsider fx as Element of (Class(E))* by FINSEQ_1:def 11;
A11: len (f*x) = len x by FINSEQ_3:120;
  now
    let m;
    assume
A12: m in dom x;
    then
A13: m in dom(f*x) by FINSEQ_3:120;
    x.m in rng x by A12,FUNCT_1:def 3;
    then reconsider xm = x.m as Element of u1;
    thus Class(E,x.m) = f.xm by Def11
      .= fx.m by A13,FINSEQ_3:120;
  end;
  then
A14: x is_representatives_FS fx by A11,FINSEQ_3:def 4;
  assume
A15: x in dom o1;
  then o1.x in rng o1 by FUNCT_1:def 3;
  then reconsider ox = o1.x as Element of u1;
  dom o1 = (arity o1)-tuples_on u1 by MARGREL1:22
    .= {p where p is Element of u1* : len p = arity o1} by FINSEQ_2:def 4;
  then
A16: ex p be Element of u1* st p = x1 & len p = arity o1 by A15;
A17: f.(o1.x) = Class(E,ox) by Def11
    .= Class(E,o1.x);
  dom QuotOp(o1,E) = (arity o1)-tuples_on Class(E) by Def8
    .= {q where q is Element of (Class(E))*: len q = arity o1} by
FINSEQ_2:def 4;
  then fx in dom QuotOp(o1,E) by A16,A11;
  hence thesis by A17,A10,A14,Def8;
end;

theorem
  for U1,E holds Nat_Hom(U1,E) is_epimorphism
proof
  let U1,E;
  set f = Nat_Hom(U1,E), qa = QuotUnivAlg(U1,E), cqa = the carrier of qa, u1 =
  the carrier of U1;
  thus f is_homomorphism by Th17;
  thus rng f c= cqa;
  let x be object;
  assume
A1: x in cqa;
  then reconsider x1 = x as Subset of u1;
  consider y being object such that
A2: y in u1 and
A3: x1 = Class(E,y) by A1,EQREL_1:def 3;
  reconsider y as Element of u1 by A2;
  dom f = u1 by FUNCT_2:def 1;
  then f.y in rng f by FUNCT_1:def 3;
  hence thesis by A3,Def11;
end;

definition
  let U1,U2;
  let f be Function of U1,U2;
  assume
A1: f is_homomorphism;
  func Cng(f) -> Congruence of U1 means
  :Def12:
  for a,b be Element of U1 holds [a,b] in it iff f.a = f.b;
  existence
  proof
    defpred P[set,set] means f.$1 = f.$2;
    set u1 = the carrier of U1;
    consider R being Relation of u1,u1 such that
A2: for x,y being Element of u1 holds [x,y] in R iff P[x,y] from
    RELSET_1:sch 2;
    R is_reflexive_in u1
    proof
      let x be object;
      assume x in u1;
      then reconsider x1 = x as Element of u1;
      f.x1 =f.x1;
      hence thesis by A2;
    end;
    then
A3: dom R = u1 & field R = u1 by ORDERS_1:13;
A4: R is_transitive_in u1
    proof
      let x,y,z be object;
      assume that
A5:   x in u1 & y in u1 & z in u1 and
A6:   [x,y] in R & [y,z] in R;
      reconsider x1 = x, y1=y, z1 = z as Element of u1 by A5;
      f.x1 = f.y1 & f.y1 = f.z1 by A2,A6;
      hence thesis by A2;
    end;
    R is_symmetric_in u1
    proof
      let x,y be object;
      assume that
A7:   x in u1 & y in u1 and
A8:   [x,y] in R;
      reconsider x1 = x, y1=y as Element of u1 by A7;
      f.x1 = f.y1 by A2,A8;
      hence thesis by A2;
    end;
    then reconsider R as Equivalence_Relation of U1 by A3,A4,PARTFUN1:def 2
,RELAT_2:def 11,def 16;
    now
      U1,U2 are_similar by A1;
      then
A9:   signature U1 = signature U2;
      let n be Nat,o be operation of U1;
      assume that
A10:  n in dom the charact of(U1) and
A11:  o = (the charact of U1).n;
      len (signature U1) = len the charact of(U1) & len (signature U2) =
      len the charact of(U2) by UNIALG_1:def 4;
      then dom the charact of(U2) = dom the charact of(U1) by A9,FINSEQ_3:29;
      then reconsider o2 = (the charact of U2).n as operation of U2 by A10,
FUNCT_1:def 3;
      let x,y be FinSequence of U1;
      assume that
A12:  x in dom o & y in dom o and
A13:  [x,y] in ExtendRel(R);
      o.x in rng o & o.y in rng o by A12,FUNCT_1:def 3;
      then reconsider ox = o.x, oy = o.y as Element of u1;
A14:  len x = len y by A13,FINSEQ_3:def 3;
A15:  len (f*y) = len y by FINSEQ_3:120;
      then
A16:  dom (f*y) = Seg len x by A14,FINSEQ_1:def 3;
A17:  len (f*x) = len x by FINSEQ_3:120;
A18:  now
        let m be Nat;
        assume
A19:    m in dom (f*y);
        then m in dom y by A14,A16,FINSEQ_1:def 3;
        then
A20:    y.m in rng y by FUNCT_1:def 3;
A21:    m in dom x by A16,A19,FINSEQ_1:def 3;
        then x.m in rng x by FUNCT_1:def 3;
        then reconsider xm = x.m, ym = y.m as Element of u1 by A20;
        [x.m,y.m] in R by A13,A21,FINSEQ_3:def 3;
        then
A22:    f.xm = f.ym by A2
          .= (f*y).m by A19,FINSEQ_3:120;
        m in dom (f*x) by A17,A16,A19,FINSEQ_1:def 3;
        hence (f*y).m = (f*x).m by A22,FINSEQ_3:120;
      end;
      f.(o.x) = o2.(f*x) & f.(o.y) = o2.(f*y) by A1,A10,A11,A12;
      then f.(ox) = f.(oy) by A14,A17,A15,A18,FINSEQ_2:9;
      hence [o.x,o.y] in R by A2;
    end;
    then reconsider R as Congruence of U1 by Def7;
    take R;
    let a,b be Element of u1;
    thus [a,b] in R implies f.a = f.b by A2;
    assume f.a = f.b;
    hence thesis by A2;
  end;
  uniqueness
  proof
    set u1 = the carrier of U1;
    let X,Y be Congruence of U1;
    assume that
A23: for a,b be Element of U1 holds [a,b] in X iff f.a = f.b and
A24: for a,b be Element of U1 holds [a,b] in Y iff f.a = f.b;
    for x,y be object holds [x,y] in X iff [x,y] in Y
    proof
      let x,y be object;
      thus [x,y] in X implies [x,y] in Y
      proof
        assume
A25:    [x,y] in X;
        then reconsider x1 = x,y1 = y as Element of u1 by ZFMISC_1:87;
        f.x1 = f.y1 by A23,A25;
        hence thesis by A24;
      end;
      assume
A26:  [x,y] in Y;
      then reconsider x1 = x,y1 = y as Element of u1 by ZFMISC_1:87;
      f.x1 = f.y1 by A24,A26;
      hence thesis by A23;
    end;
    hence thesis by RELAT_1:def 2;
  end;
end;

definition
  let U1,U2 be Universal_Algebra, f be Function of U1,U2;
  assume
A1: f is_homomorphism;
  func HomQuot(f) -> Function of QuotUnivAlg(U1,Cng(f)),U2 means
  :Def13:
  for a be Element of U1 holds it.(Class(Cng f,a)) = f.a;
  existence
  proof
    set qa = QuotUnivAlg(U1,Cng(f)), cqa = the carrier of qa, u1 = the carrier
    of U1, u2 = the carrier of U2;
    defpred P[object,object] means
   for a be Element of u1 st $1 = Class(Cng f,a)
    holds $2 = f.a;
A2: for x being object st x in cqa ex y being object st y in u2 & P[x,y]
    proof
      let x be object;
      assume
A3:   x in cqa;
      then reconsider x1 = x as Subset of u1;
      consider a be object such that
A4:   a in u1 and
A5:   x1 = Class(Cng f,a) by A3,EQREL_1:def 3;
      reconsider a as Element of u1 by A4;
      take y = f.a;
      thus y in u2;
      let b be Element of u1;
      assume x = Class(Cng f,b);
      then b in Class(Cng f,a) by A5,EQREL_1:23;
      then [b,a] in Cng(f) by EQREL_1:19;
      hence thesis by A1,Def12;
    end;
    consider F being Function such that
A6: dom F = cqa & rng F c= u2 & for x being object st x in cqa holds P[x,F.x]
     from
    FUNCT_1:sch 6(A2);
    reconsider F as Function of qa,U2 by A6,FUNCT_2:def 1,RELSET_1:4;
    take F;
    let a be Element of u1;
    Class(Cng f,a) in Class(Cng f) by EQREL_1:def 3;
    hence thesis by A6;
  end;
  uniqueness
  proof
    set qa = QuotUnivAlg(U1,Cng(f)), cqa = the carrier of qa, u1 = the carrier
    of U1;
    let F,G be Function of qa,U2;
    assume that
A7: for a be Element of u1 holds F.(Class(Cng f,a)) = f.a and
A8: for a be Element of u1 holds G.(Class(Cng f,a)) = f.a;
    let x be Element of cqa;
    x in cqa;
    then reconsider x1 = x as Subset of u1;
    consider a be object such that
A9: a in u1 & x1 = Class(Cng f,a) by EQREL_1:def 3;
    thus F.x = f.a by A7,A9
      .= G.x by A8,A9;
  end;
end;

theorem Th19:
  f is_homomorphism implies HomQuot(f) is_homomorphism
& HomQuot(f) is_monomorphism
proof
  set qa = QuotUnivAlg(U1,Cng(f)), cqa = the carrier of qa, u1 = the carrier
  of U1, F = HomQuot(f);
  assume
A1: f is_homomorphism;
  thus
A2: F is_homomorphism
  proof
    Nat_Hom(U1,Cng f) is_homomorphism by Th17;
    then U1,qa are_similar;
    then
A3: signature U1 = signature qa;
    U1,U2 are_similar by A1;
    then signature U2 = signature qa by A3;
    hence qa,U2 are_similar;
    let n;
    assume
A4: n in dom the charact of(qa);
A5: len (signature U1) = len the charact of(U1) & len (signature qa) =
    len the charact of(qa) by UNIALG_1:def 4;
A6: dom the charact of(qa) = Seg len (the charact of qa) & dom the
    charact of(U1 ) = Seg len (the charact of U1) by FINSEQ_1:def 3;
    then reconsider o1 = (the charact of U1).n as operation of U1 by A3,A4,A5,
FUNCT_1:def 3;
A7: dom o1 = (arity o1)-tuples_on u1 by MARGREL1:22
      .= {p where p is Element of u1* : len p = arity o1} by FINSEQ_2:def 4;
    let oq be operation of qa, o2 be operation of U2;
    assume that
A8: oq = (the charact of qa).n and
A9: o2 = (the charact of U2).n;
    let x be FinSequence of qa;
    assume
A10: x in dom oq;
    reconsider x1 = x as FinSequence of Class(Cng f);
    reconsider x1 as Element of (Class(Cng f))* by FINSEQ_1:def 11;
    consider y be FinSequence of U1 such that
A11: y is_representatives_FS x1 by FINSEQ_3:122;
    reconsider y as Element of u1* by FINSEQ_1:def 11;
A12: len x1 = len y by A11,FINSEQ_3:def 4;
    then
A13: len (F*x) = len y by FINSEQ_3:120;
A14: len y = len (f*y) by FINSEQ_3:120;
A15: now
      let m be Nat;
      assume
A16:  m in Seg len y;
      then
A17:  m in dom (F*x) by A13,FINSEQ_1:def 3;
A18:  m in dom(f*y) by A14,A16,FINSEQ_1:def 3;
A19:  m in dom y by A16,FINSEQ_1:def 3;
      then reconsider ym = y.m as Element of u1 by FINSEQ_2:11;
      x1.m = Class(Cng f,y.m) by A11,A19,FINSEQ_3:def 4;
      hence (F*x).m = F.(Class(Cng f,ym)) by A17,FINSEQ_3:120
        .= f.(y.m) by A1,Def13
        .= (f*y).m by A18,FINSEQ_3:120;
    end;
    dom(F*x) = Seg len y by A13,FINSEQ_1:def 3;
    then
A20: o2.(F*x) = o2.(f*y) by A13,A14,A15,FINSEQ_2:9;
A21: oq = QuotOp(o1,Cng f) by A4,A8,Def9;
    then dom oq = (arity o1)-tuples_on Class(Cng f) by Def8
      .= {w where w is Element of (Class(Cng f))*: len w = arity o1} by
FINSEQ_2:def 4;
    then ex w be Element of (Class(Cng f))* st w = x1 & len w = arity o1 by A10
;
    then
A22: y in dom o1 by A12,A7;
    then o1.y in rng o1 by FUNCT_1:def 3;
    then reconsider o1y = o1.y as Element of u1;
    F.(oq.x) = F.(Class(Cng f,o1y)) by A10,A11,A21,Def8
      .= f.(o1.y) by A1,Def13;
    hence thesis by A1,A3,A4,A9,A6,A5,A22,A20;
  end;
A23: dom F = cqa by FUNCT_2:def 1;
  F is one-to-one
  proof
    let x,y be object;
    assume that
A24: x in dom F and
A25: y in dom F and
A26: F.x = F.y;
    reconsider x1 = x, y1 = y as Subset of u1 by A23,A24,A25;
    consider a be object such that
A27: a in u1 and
A28: x1 = Class(Cng f,a) by A24,EQREL_1:def 3;
    reconsider a as Element of u1 by A27;
    consider b be object such that
A29: b in u1 and
A30: y1 = Class(Cng f,b) by A25,EQREL_1:def 3;
    reconsider b as Element of u1 by A29;
A31: F.y1 = f.b by A1,A30,Def13;
    F.x1 = f.a by A1,A28,Def13;
    then [a,b] in Cng(f) by A1,A26,A31,Def12;
    hence thesis by A28,A30,EQREL_1:35;
  end;
  hence thesis by A2;
end;

::$N First isomorphism theorem for universal algebras
theorem Th20:
  f is_epimorphism implies HomQuot(f) is_isomorphism
proof
  set qa = QuotUnivAlg(U1,Cng(f)), u1 = the carrier of U1, u2 = the carrier of
  U2, F = HomQuot(f);
  assume
A1: f is_epimorphism;
  then
A2: f is_homomorphism;
  then F is_monomorphism by Th19;
  then
A3: F is one-to-one;
A4: rng f = u2 by A1;
A5: rng F = u2
  proof
    thus rng F c= u2;
    let x be object;
    assume x in u2;
    then consider y being object such that
A6: y in dom f and
A7: f.y = x by A4,FUNCT_1:def 3;
    reconsider y as Element of u1 by A6;
    set u = Class(Cng f,y);
A8: dom F = the carrier of qa & u in Class(Cng f) by EQREL_1:def 3
,FUNCT_2:def 1;
    F.u = x by A2,A7,Def13;
    hence thesis by A8,FUNCT_1:def 3;
  end;
  F is_homomorphism by A2,Th19;
  hence thesis by A3,A5,Th7;
end;

theorem
  f is_epimorphism implies QuotUnivAlg(U1,Cng(f)),U2 are_isomorphic
proof
  assume
A1: f is_epimorphism;
  take HomQuot(f);
  thus thesis by A1,Th20;
end;