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 Th36:
  for H2 being strict StableSubgroup of G holds H1 is
  StableSubgroup of H2 iff H1 "\/" H2 = H2
proof
  let H2 be strict StableSubgroup of G;
  thus H1 is StableSubgroup of H2 implies H1 "\/" H2 = H2
  proof
    assume H1 is StableSubgroup of H2;
    then H1 is Subgroup of H2 by Def7;
    then the carrier of H1 c= the carrier of H2 by GROUP_2:def 5;
    hence H1 "\/" H2 = the_stable_subgroup_of carr H2 by XBOOLE_1:12
      .= H2 by Th25;
  end;
  thus thesis by Th35;
end;
