theorem Th126:
       L is vf-qc-correct subst-correct implies
  (\for(x,A)\and\for(x,B))\imp\for(x,A\andB) in G
  proof set Y = X extended_by ({},the carrier of S1);
    assume
A1: L is vf-qc-correct subst-correct;
    then \for(x,A)\impA in G & \for(x,B)\impB in G by Th104;
    then \for(x,A)\and\for(x,B)\imp(A\andB) in G by Th72;
    then
A2: \for(x,\for(x,A)\and\for(x,B)\imp(A\andB)) 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: dom X = the carrier of J c= the carrier of S1 = dom Y
    by INSTALG1:10,PARTFUN1:def 2;
    reconsider a as SortSymbol of J by A3;
    x nin (vf \for(x,A)).a & x nin (vf \for(x,B)).a by A1,A3,A4,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))).a by A4,PBOOLE:def 4;
    then x nin ((vf (\for(x,A)\and\for(x,B)))).a by A1;
    hence (\for(x,A)\and\for(x,B))\imp\for(x,A\andB) in G by A3,A2,Th108;
  end;
