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 Th29:
  H1 "\/" H2 = the_stable_subgroup_of(H1 * H2)
proof
  reconsider H19=H1,H29=H2 as Subgroup of G by Def7;
  reconsider Y = the carrier of (H19"\/"H29) as Subset of G by GROUP_2:def 5;
A1: Y = the carrier of gr(H19*H29) by GROUP_4:50;
  H1 "\/" H2 = the_stable_subgroup_of Y by Lm31
    .= the_stable_subgroup_of(H19*H29) by A1,Lm31;
  hence thesis;
end;
