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));

theorem Th110:
       L is subst-correct implies
  for a being SortSymbol of J st x in X.a & y in X.a & x0 = x & y0 = y
  holds A/(x0,y0)\imp\ex(x,A) in G
  proof set Y = X extended_by ({},the carrier of S1);
    assume
A1: L is subst-correct;
    let a be SortSymbol of J such that
A2: x in X.a & y in X.a & x0 = x & y0 = y;
    J is Subsignature of S1 by Def2;
    then the carrier of J c= the carrier of S1 by INSTALG1:10;
    then
A3: a in the carrier of S1 & X c= the Sorts of T &
    dom the Sorts of L = the carrier of S1 by PARTFUN1:def 2,PBOOLE:def 18;
    then (the Sorts of L).a in rng the Sorts of L &
    (the Sorts of L).a = (the Sorts of T).a by Th16,FUNCT_1:def 3;
    then
A4: X.a c= (the Sorts of T).a = (the Sorts of L).a
    c= Union the Sorts of L by A3,ZFMISC_1:74;
    then reconsider t = y as Element of Union the Sorts of L by A2;
A5: a is SortSymbol of S1 by Th8; then
A6: x in Y.a & y in Y.a by A2,Th2;
A7: Y is ManySortedSubset of the Sorts of L by Th23;
    \for(x,\notA)\imp((\notA)/(x0,t)) in G by A2,A4,A6,Def39;
    then \for(x,\notA)\imp((\notA)/(x0,y0)) in G by A2,A6,A7,A5,Th14;
    then \not((\notA)/(x0,y0))\imp\not\for(x,\notA) in G by Th58;
    then \not\not(A/(x0,y0))\imp\not\for(x,\notA) in G &
    (A/(x0,y0))\imp\not\not(A/(x0,y0)) in G &
    \ex(x,A)\iff\not\for(x,\notA) in G &
    \ex(x,A)\iff\not\for(x,\notA)\imp(\not\for(x,\notA)\imp\ex(x,A)) in G
    by A1,A2,A6,A7,A5,Th27,Def38,Th64,Th105;
    then (A/(x0,y0))\imp\not\for(x,\notA) in G &
    \not\for(x,\notA)\imp\ex(x,A) in G by Th45,Def38;
    hence thesis by Th45;
  end;
