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;
