theorem
  [.H1, (Omega).G.] is Subgroup of H2
  implies [.H1 /\ H,H.] is Subgroup of H2 /\ H
proof
  assume
A1: [.H1, (Omega).G.] is Subgroup of H2;
  H1 /\ H is Subgroup of H by GROUP_2:88;
  then
A2:[.H1 /\ H,H.] is Subgroup of H by Th11;
A3: H is Subgroup of (Omega).G by Lm2;
  H1 /\ H is Subgroup of H1 by GROUP_2:88;
  then [.H1 /\ H,H.] is Subgroup of [.H1, (Omega).G.] by A3,GROUP_5:66;
  then [.H1 /\ H,H.] is Subgroup of H2 by A1,GROUP_2:56;
  hence thesis by A2,GROUP_2:91;
end;
