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 Th109:
       L is subst-correct vf-qc-correct implies
  \for(x,A\impB)\imp(\for(x,A)\imp\for(x,B)) in G
  proof set Y = X extended_by ({},the carrier of S1);
    consider a being object such that
A1: a in dom X & x in X.a by CARD_5:2;
    J is Subsignature of S1 by Def2;
    then dom X = the carrier of J c= the carrier of S1 = dom Y
    by INSTALG1:10,PARTFUN1:def 2;
    then reconsider a as SortSymbol of S1 by A1;
    assume
A2: L is subst-correct vf-qc-correct;
    then \for(x,A\impB)\imp(\for(x,A)\impB) in G by Th107;
    then
A3: \for(x,\for(x,A\impB)\imp(\for(x,A)\impB)) in G by Def39;
A4: vf \for(x,A\impB) = vf(A\impB) (\) a-singleton x & x in {x}
    by A1,A2,TARSKI:def 1;
    then (vf \for(x,A\impB)).a
    = (vf(A\impB)).a \ (a-singleton x).a by PBOOLE:def 6
    .= (vf(A\impB)).a \ {x} by AOFA_A00:6;
    then x nin (vf \for(x,A\impB)).a by A4,XBOOLE_0:def 5;
    then
A5: \for(x,A\impB)\imp\for(x,\for(x,A)\impB) in G by A1,A3,Th108;
A6: vf \for(x,A) = vf A (\) a-singleton x & x in {x}
    by A1,A2,TARSKI:def 1;
    then (vf \for(x,A)).a = (vf(A)).a \ (a-singleton x).a by PBOOLE:def 6
    .= (vf(A)).a \ {x} by AOFA_A00:6;
    then x nin (vf \for(x,A)).a by A6,XBOOLE_0:def 5;
    then \for(x,\for(x,A)\impB)\imp(\for(x,A)\imp\for(x,B)) in G by A1,Def39;
    hence thesis by A5,Th45;
  end;
