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 Th134:
  L is subst-correct vf-qc-correct &
  x in X.a & x nin (vf A).a implies \ex(x,A\andB)\impA\and\ex(x,B) in G
  proof
    assume
A1: L is subst-correct vf-qc-correct;
    assume
A2: x in X.a & x nin (vf A).a;
    (\for(x,\notA)\or\for(x,\notB))\imp\for(x,\notA\or\notB) in G by A1,Th127;
    then
A3: \not\for(x,\notA\or\notB)\imp\not(\for(x,\notA)\or\for(x,\notB)) in G
    by Th58;
    \notA\or\notB\imp\not(A\andB) in G by Th73;
    then \ex(x,A\andB)\iff\not\for(x,\not(A\andB)) in G &
    \for(x,\notA\or\notB)\imp\for(x,\not(A\andB)) in G by A1,Th115,Th105;
    then \not\for(x,\not(A\andB))\imp\not\for(x,\notA\or\notB) in G &
    \ex(x,A\andB)\imp\not\for(x,\not(A\andB)) in G by Th43,Th58;
    then \ex(x,A\andB)\imp\not\for(x,\notA\or\notB) in G by Th45;
    then
A4: \ex(x,A\andB)\imp\not(\for(x,\notA)\or\for(x,\notB)) in G by A3,Th45;
    \ex(x,A)\iff\not\for(x,\notA) in G & \ex(x,B)\iff\not\for(x,\notB) in G
    by Th105;
    then \not\for(x,\notA)\imp\ex(x,A) in G &
    \not\for(x,\notB)\imp\ex(x,B) in G by Th43;
    then
A5: \not(\for(x,\notA))\and\not\for(x,\notB)\imp\ex(x,A)\and\ex(x,B) in G
    by Th72;
    \not(\for(x,\notA)\or\for(x,\notB))\imp
    \not\for(x,\notA)\and\not\for(x,\notB) in G by Th71;
    then \ex(x,A\andB)\imp\not\for(x,\notA)\and\not\for(x,\notB) in G
    by A4,Th45;
    then
A6: \ex(x,A\andB)\imp\ex(x,A)\and\ex(x,B) in G by A5,Th45;
    A\impA in G by Th34;
    then \for(x,A\impA) in G &
    \for(x,A\impA)\imp(\ex(x,A)\impA) in G by A1,A2,Th120,Def39;
    then \ex(x,A)\impA in G & \ex(x,B)\imp\ex(x,B) in G by Def38,Th34;
    then \ex(x,A)\and\ex(x,B)\impA\and\ex(x,B) in G by Th72;
    hence \ex(x,A\andB)\impA\and\ex(x,B) in G by A6,Th45;
  end;
