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 ThTwo:
  L is subst-correct & x = x0 in X.s & y = y0 in X.s implies
  A\and (x '=' (y,L))\imp(A/(x0,y0)) in G1
  proof
    assume that
A0: L is subst-correct and
A1: x = x0 in X.s & y = y0 in X.s;
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);
    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;
A3: t1 '=' (t2,L)\imp(A/(x0,t1)\imp(A/(x0,t2))) in G1 by A1,Def42;
A5: A/(x0,t1) = A/(x0,x0) by A1,A4,Th14 .= A by A0,A1,A4;
    A/(x0,t2) = A/(x0,y0) by A1,A4,Th14;
    then A\imp(t1 '=' (t2,L)\imp(A/(x0,y0))) in G1 &
    (A\imp(t1 '=' (t2,L)\imp(A/(x0,y0))))\imp
    (A\and(t1 '=' (t2,L))\imp(A/(x0,y0))) in G1 by A3,A5,Th38,Th48;
    hence thesis by Def38;
  end;
