reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

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;
