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 Th127:
  L is subst-correct vf-qc-correct implies
  (\for(x,A)\or\for(x,B))\imp\for(x,A\orB) in G
  proof set Y = X extended_by ({},the carrier of S1);
    assume
A1: L is subst-correct vf-qc-correct;
    then \for(x,A)\impA in G & \for(x,B)\impB in G by Th104;
    then \for(x,A)\or\for(x,B)\imp(A\orB) in G by Th59;
    then
A2: \for(x,\for(x,A)\or\for(x,B)\imp(A\orB)) in G by Def39;
    consider a being object such that
A3: a in dom X & x in X.a by CARD_5:2;
    J is Subsignature of S1 by Def2;
    then
A4: the carrier of J c= the carrier of S1 = dom Y &
    dom X = the carrier of J by PARTFUN1:def 2,INSTALG1:10;
    reconsider a as SortSymbol of J by A3;
    reconsider b = a as SortSymbol of S1 by A4;
    x nin (vf \for(x,A)).b & x nin (vf \for(x,B)).b by A1,A3,Th113;
    then x nin (vf \for(x,A)).a \/ (vf \for(x,B)).a by XBOOLE_0:def 3;
    then x nin ((vf \for(x,A)) (\/) (vf \for(x,B))).b by PBOOLE:def 4;
    then x nin ((vf (\for(x,A)\or\for(x,B)))).a by A1;
    hence (\for(x,A)\or\for(x,B))\imp\for(x,A\orB) in G by A2,A3,Th108;
  end;
