reserve i for Element of NAT;

theorem Th1:
  for G being Group, H,F1,F2 being strict Subgroup of G st
  F1 is normal Subgroup of F2 holds F1 /\ H is normal Subgroup of F2 /\ H
proof
  let G be Group;
  let H,F1,F2 be strict Subgroup of G;
  reconsider F=F2 /\ H as Subgroup of F2 by GROUP_2:88;
  assume
A1: F1 is normal Subgroup of F2;
  then
A2: (F1 /\ H) = ((F1 /\ F2) /\ H) by GROUP_2:89
    .= F1 /\ (F2 /\ H) by GROUP_2:84;
  reconsider F1 as normal Subgroup of F2 by A1;
  F1/\F is normal Subgroup of F;
  hence thesis by A2,GROUP_6:3;
end;
