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));
reserve a for SortSymbol of J;

theorem
  L is subst-correct vf-qc-correct implies
  \for(x,A\andA)\iff\for(x,A) in G
  proof
    assume A1: L is subst-correct vf-qc-correct;
    A\andA\impA in G & A\impA\andA in G by Def38,Th53;
    then
A2: \for(x,A\andA\impA) in G & \for(x,A\impA\andA) in G by Def39;
    \for(x,A\impA\andA)\imp(\for(x,A)\imp\for(x,A\andA)) in G &
    \for(x,A\andA\impA)\imp(\for(x,A\andA)\imp\for(x,A)) in G by A1,Th109;
    then (\for(x,A)\imp\for(x,A\andA)) in G &
    (\for(x,A\andA)\imp\for(x,A)) in G by A2,Def38;
    hence \for(x,A\andA)\iff\for(x,A) in G by Th43;
  end;
