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 &
  x in X.a & x nin (vf A).a implies \ex(x,A\andB)\iffA\and\ex(x,B) in G
  proof set Y = X extended_by ({},the carrier of S1);
    assume A1: L is subst-correct vf-qc-correct;
    assume A2: x in X.a;
    assume A3: x nin (vf A).a;
    vf \notA = vf A by A1;
    then \for(x,\notA\or\notB)\imp\notA\or\for(x,\notB) in G by A1,A2,A3,Th133;
    then
A4: \not(\notA\or\for(x,\notB))\imp\not\for(x,\notA\or\notB) in G by Th58;
    \for(x,\notB)\iff\not\ex(x,B) in G by A1,Th122;
    then \notA\imp\notA in G & \for(x,\notB)\imp\not\ex(x,B) in G by Th34,Th43;
    then \notA\or\not\ex(x,B)\imp\not(A\and\ex(x,B)) in G &
    (\notA\or\for(x,\notB))\imp(\notA\or\not\ex(x,B)) in G &
    (\notA\or\not\ex(x,B))\imp\not(A\and\ex(x,B)) in G by Th59,Th73;
    then (A\and\ex(x,B))\imp\not(\notA\or\not\ex(x,B)) in G &
    \not(\notA\or\not\ex(x,B))\imp\not(\notA\or\for(x,\notB)) in G
    by Th58,Th67;
    then (A\and\ex(x,B))\imp\not(\notA\or\for(x,\notB)) in G by Th45;
    then
A5: A\and\ex(x,B)\imp\not\for(x,\notA\or\notB) in G by A4,Th45;
    \ex(x,A\andB)\iff\not\for(x,\not(A\andB)) in G &
    \not(A\andB)\imp\notA\or\notB in G by Th70,Th105;
    then \not\for(x,\not(A\andB))\imp\ex(x,A\andB) in G &
    \for(x,\not(A\andB)\imp\notA\or\notB) in G &
    \for(x,\not(A\andB)\imp\notA\or\notB)\imp
    (\for(x,\not(A\andB))\imp\for(x,\notA\or\notB)) in G
    by A1,Def39,Th43,Th109;
    then \not\ex(x,A\andB)\imp\for(x,\not(A\andB)) in G &
    \for(x,\not(A\andB))\imp\for(x,\notA\or\notB) in G by Def38,Th68;
    then \not\ex(x,A\andB)\imp\for(x,\notA\or\notB) in G by Th45;
    then \not\for(x,\notA\or\notB)\imp\ex(x,A\andB) in G by Th68;
    then
A6: A\and\ex(x,B)\imp\ex(x,A\andB) in G by A5,Th45;
    \ex(x,A\andB)\impA\and\ex(x,B) in G by Th134,A1,A2,A3;
    hence \ex(x,A\andB)\iffA\and\ex(x,B) in G by A6,Th43;
  end;
