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 Th112:
  L is subst-correct implies A\imp\ex(x,A) in G
  proof set Y = X extended_by ({},the carrier of S1);
    assume
A1: L is subst-correct;
    consider a being object such that
A2: a in dom X & x in X.a by CARD_5:2;
    reconsider a as SortSymbol of J by A2;
A3: x in X.a & a is SortSymbol of S1 by A2,Th8; then
A4: x in Y.a & dom Y = the carrier of S1 by Th2,PARTFUN1:def 2;
    then reconsider x0 = x as Element of Union Y by A3,CARD_5:2;
    A/(x0,x0)\imp\ex(x,A) in G by A1,A2,Th110;
    hence thesis by A1,A3,A4;
  end;
