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 Th133:
  L is subst-correct vf-qc-correct &
  x in X.a & x nin (vf A).a implies \for(x,A\orB)\imp(A\or\for(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;
    set c = a, a = \notA, b = B;
    x nin (vf a).c by A1,A2; then
A3: \for(x,a\impb)\imp(a\imp\for(x,b)) in G by A2,Def39;
    A\orb\imp(a\impb) in G by Th62;
    then \for(x,A\orB)\imp\for(x,a\impB) in G by A1,Th115;
    then
A4: \for(x,A\orB)\imp(a\imp\for(x,b)) in G by A3,Th45;
    \nota\impA in G & \for(x,B)\imp\for(x,B) in G by Th34,Th65;
    then \nota\or\for(x,B)\impA\or\for(x,B) in G &
    a\imp\for(x,B)\imp\nota\or\for(x,B) in G by Th59,Th82;
    then a\imp\for(x,B)\impA\or\for(x,B) in G by Th45;
    hence thesis by A4,Th45;
  end;
