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 Th113:
  L is vf-qc-correct implies
  for a being SortSymbol of S1 st x in X.a
  holds x nin (vf \for(x,A)).a
  proof set Y = X extended_by ({},the carrier of S1);
    assume
A1: L is vf-qc-correct;
    let a be SortSymbol of S1; assume x in X.a;
    then vf \for(x,A) = (vf A)(\)(a-singleton x) by A1;
    then
A2: (vf \for(x,A)).a = ((vf A).a) \ ((a-singleton x).a) by PBOOLE:def 6
    .= (vf A).a \ {x} by AOFA_A00:6;
    x in {x} by TARSKI:def 1;
    hence thesis by A2,XBOOLE_0:def 5;
  end;
