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
  L is subst-correct implies
  for a being SortSymbol of J st x in X.a & x nin (vf A).a
  holds A\iff\for(x,A) in G
  proof
    assume
A1: L is subst-correct;
    let a be SortSymbol of J; assume
A2: x in X.a & x nin (vf A).a;
A3: \for(x,A\impA)\imp(\for(x,A)\impA) in G by A1,Th107;
    A\impA in G by Th34;
    then \for(x,A\impA) in G by Def39;
    then \for(x,A)\impA in G & A\imp\for(x,A) in G by A2,A3,Def38,Th108;
    hence thesis by Th43;
  end;
