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 Th104:
  L is subst-correct implies \for(x,A)\impA 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;
    J is Subsignature of S1 by Def2;
    then
A3: the carrier of J c= the carrier of S1 & dom Y = the carrier of S1 &
    dom X = the carrier of J by PARTFUN1:def 2,INSTALG1:10;
    then reconsider a as SortSymbol of S1 by A2;
A4: x in Y.a by A2,A3,Th1;
    then reconsider x0 = x as Element of Union Y by A3,CARD_5:2;
    X c= the Sorts of T by PBOOLE:def 18;
    then X.a c= (the Sorts of T).a = (the Sorts of L).a
    by A2,Th16;
    then reconsider t = x as Element of (the Sorts of L).a by A2;
    Y is ManySortedSubset of the Sorts of L by Th23;
    then A/(x0,t) = A/(x0,x0) by A4,Th14 .= A by A4,A1;
    hence thesis by A2,A4,Def39;
  end;
