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 & x = x0 in X.s & y = y0 in X.s implies
  A\and\not(A/(x0,y0))\imp\not(x '=' (y,L)) in G1
  proof
    assume that
A0: L is subst-correct and
A1: x = x0 in X.s & y = y0 in X.s;
    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;
    A\and (x '=' (y,L))\imp(A/(x0,y0)) in G1 &
    (A\and(t1 '=' (t2,L))\imp(A/(x0,y0)))\imp
    (A\imp((t1 '=' (t2,L))\imp(A/(x0,y0)))) in G1 by A0,A1,ThTwo,Th47;
    then
    A\imp((t1 '=' (t2,L))\imp(A/(x0,y0))) in G1 &
   ((t1 '=' (t2,L))\imp(A/(x0,y0)))\imp(\not(A/(x0,y0))\imp\not(t1 '=' (t2,L)))
   in G1 by Def38,Th57;
   then
   A\imp(\not(A/(x0,y0))\imp\not(t1 '=' (t2,L))) in G1 &
   A\imp(\not(A/(x0,y0))\imp\not(t1 '=' (t2,L)))\imp
   (A\and\not(A/(x0,y0))\imp\not(t1 '=' (t2,L))) in G1 by Th45,Th48;
   hence thesis by Def38;
  end;
