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 Th1:
 for x being object holds x in H1 implies x in G
proof let x be object;
  assume
A1: x in H1;
  H1 is Subgroup of G by Def7;
  hence thesis by A1,GROUP_2:40;
end;
