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 Th13:
  (for g being Element of G st g in H1 holds g in H2) implies H1
  is StableSubgroup of H2
proof
  assume
A1: for g being Element of G st g in H1 holds g in H2;
  the carrier of H1 c= the carrier of H2
  proof
    let x be object;
    assume x in the carrier of H1;
    then reconsider g = x as Element of H1;
    reconsider g as Element of G by Th2;
    g in H1 by STRUCT_0:def 5;
    then g in H2 by A1;
    hence thesis by STRUCT_0:def 5;
  end;
  hence thesis by Lm20;
end;
