reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;

theorem Th60:
  for H,F1,F2 being strict StableSubgroup of G st F1 is normal
  StableSubgroup of F2 holds H /\ F1 is normal StableSubgroup of H /\ F2
proof
  let H,F1,F2 be strict StableSubgroup of G;
  reconsider F=F2 /\ H as StableSubgroup of F2 by Lm33;
  assume
A1: F1 is normal StableSubgroup of F2;
  then
A2: F1 /\ H=(F1 /\ F2) /\ H by Lm21
    .=F1 /\ (F2 /\ H) by Th20;
  reconsider F1 as normal StableSubgroup of F2 by A1;
  F1 /\ F is normal StableSubgroup of F by Th41;
  hence thesis by A2,Th39;
end;
