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 vf-qc-correct implies
  \ex(x,A)\or\ex(x,B)\iff\ex(x,A\orB) in G
  proof assume
A1: L is subst-correct vf-qc-correct;
    then \for(x,\notA\and\notB)\imp\for(x,\notA)\and\for(x,\notB) in G
    by Th125;
    then
A2: \not(\for(x,\notA)\and\for(x,\notB))\imp\not\for(x,\notA\and\notB)
    in G by Th58;
    \not\for(x,\notA)\or\not\for(x,\notB)\imp
    \not(\for(x,\notA)\and\for(x,\notB)) in G by Th73;
    then
A3: \not\for(x,\notA)\or\not\for(x,\notB)\imp\not\for(x,\notA\and\notB)
    in G by A2,Th45;
    \not(A\orB)\imp(\notA\and\notB) in G by Th71;
    then \for(x,\not(A\orB))\imp\for(x,\notA\and\notB) in G by A1,Th115;
    then \not\for(x,\notA\and\notB)\imp\not\for(x,\not(A\orB)) in G by Th58;
    then
A4: \not\for(x,\notA)\or\not\for(x,\notB)\imp\not\for(x,\not(A\orB)) in G
    by A3,Th45;
    \ex(x,A\orB)\iff\not\for(x,\not(A\orB)) in G by Th105;
    then \not\for(x,\not(A\orB))\iff\ex(x,A\orB) in G by Th90;
    then
A5: \not\for(x,\notA)\or\not\for(x,\notB)\imp\ex(x,A\orB) in G by A4,Th93;
    \ex(x,A)\iff\not\for(x,\notA) in G & \ex(x,B)\iff\not\for(x,\notB) in G
    by Th105;
    then \ex(x,A)\imp\not\for(x,\notA) in G &
    \ex(x,B)\imp\not\for(x,\notB) in G by Th43;
    then \ex(x,A)\or\ex(x,B)\imp(\not\for(x,\notA)\or\not\for(x,\notB)) in G
    by Th59;
    then
A6: \ex(x,A)\or\ex(x,B)\imp\ex(x,A\orB) in G by A5,Th45;
A7: \ex(x,A)\iff\not\for(x,\notA) in G & \ex(x,B)\iff\not\for(x,\notB) in G
    by Th105;
A8: \not\for(x,\notA)\imp\ex(x,A) in G &
    \not\for(x,\notB)\imp\ex(x,B) in G by A7,Th43;
    \ex(x,A\orB)\iff\not\for(x,\not(A\orB)) in G by Th105;
    then
A9: \ex(x,A\orB)\imp\not\for(x,\not(A\orB)) in G by Th43;
    \notA\and\notB\imp\not(A\orB) in G by Th74;
    then \for(x,\notA\and\notB)\imp\for(x,\not(A\orB)) in G by A1,Th115;
    then \not\for(x,\not(A\orB))\imp\not\for(x,\notA\and\notB) in G by Th58;
    then
A10: \ex(x,A\orB)\imp\not\for(x,\notA\and\notB) in G by A9,Th45;
A11: \not\for(x,\notA)\or\not\for(x,\notB)\imp\ex(x,A)\or\ex(x,B) in G
    by A8,Th59;
    \not(\for(x,\notA)\and\for(x,\notB))\imp\not\for(x,\notA)\or
    \not\for(x,\notB) in G by Th70;
    then
A12: \not(\for(x,\notA)\and\for(x,\notB))\imp\ex(x,A)\or\ex(x,B) in G
    by A11,Th45;
    (\for(x,\notA)\and\for(x,\notB))\imp\for(x,\notA\and\notB) in G
    by A1,Th126;
    then \not\for(x,\notA\and\notB)\imp\not(\for(x,\notA)\and\for(x,\notB))
    in G by Th58;
    then \ex(x,A\orB)\imp\not(\for(x,\notA)\and\for(x,\notB)) in G by A10,Th45;
    then \ex(x,A\orB)\imp\ex(x,A)\or\ex(x,B) in G by A12,Th45;
    hence thesis by A6,Th43;
  end;
