reserve X,Y for set, x,y,z for object, i,j,n for natural number;
reserve
  n for non empty Nat,
  S for non empty non void n PC-correct PCLangSignature,
  L for language MSAlgebra over S,
  F for PC-theory of L,
  A,B,C,D for Formula of L;
reserve
  J for non empty non void Signature,
  T for non-empty MSAlgebra over J,
  X for non empty-yielding GeneratorSet of T,
  S1 for J-extension non empty non void n PC-correct QC-correct
  QCLangSignature over Union X,
  L for non-empty Language of X extended_by ({},the carrier of S1), S1,
  G for QC-theory of L,
  A,B,C,D for Formula of L;
reserve x,y,z for Element of Union X;
reserve x0,y0,z0 for Element of Union (X extended_by ({},the carrier of S1));
reserve a for SortSymbol of J;
reserve
  L for
    non-empty T-extension Language of X extended_by ({},the carrier of S1), S1,
  G for QC-theory of L,
  G1 for QC-theory_with_equality of L,
  A,B,C,D for Formula of L,
  s,s1 for SortSymbol of S1,
  t,t9 for Element of L,s,
  t1,t2,t3 for Element of L,s1;

theorem
  L is subst-correct subst-eq-correct subst-correct3 vf-eq-correct &
  x in X.s & y in X.s & x <> y implies \for(x,\ex(y,x '=' (y,L))) in G1
  proof
    assume that
A0: L is subst-correct subst-eq-correct subst-correct3 vf-eq-correct and
A1: x in X.s & y in X.s & x <> y;
A2: s in dom X = the carrier of J by A1,FUNCT_1:def 2,PARTFUN1:def 2;
    then X.s c= (the Sorts of T).s = (the Sorts of L).s
    by Th16,PBOOLE:def 2,def 18;
    then reconsider t1 = x, t2 = y as Element of L,s by A1;
    reconsider j = s as SortSymbol of J by A2;
    reconsider q1 = t1, q2 = t2 as Element of T,j by Th16;
    set Y = X extended_by({}, the carrier of S1);
    reconsider y0 = y, x0 = x as Element of Union Y by Th24;
    dom Y = the carrier of S1 by PARTFUN1:def 2;
    then
A4: X.s = Y.s & (the Sorts of L).the formula-sort of S1 <> {} &
    Y is ManySortedSubset of the Sorts of L by A2,Th1,Th23;
    vf t1 = s-singleton x0 by A0,A1;
    then (vf t1).s = {x0} by AOFA_A00:6;
    then
B2: y0 nin (vf t1).s by A1,TARSKI:def 1;
A3: t1 '=' (t2,L)/(y0,x0)\imp\ex(y,t1 '=' (t2,L)) in G1 by A0,A1,A2,Th110;
A5: t1 '=' (t2,L)/(y0,x0)
    = ((t1 '=' (t2,L))/(y0,t1)) by A1,A4,Th14
    .= t1/(y0,t1) '=' (t2/(y0,t1),L) by A0,A1
    .= t1 '=' (t2/(y0,t1),L) by B2,A0,A1,A4
    .= t1 '=' (t1,L) by A0,A1,A4;
    q1 '=' (q1,L) in G1 by Def42;
    then \ex(y,x '=' (y,L)) in G1 by A3,A5,Def38;
    hence thesis by Def39;
  end;
