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 Th21:
  (1).G /\ H1 = (1).G & H1 /\ (1).G = (1).G
proof
A1: (1).G is StableSubgroup of H1 by Th16;
  hence (1).G /\ H1 = (1).G by Lm21;
  thus thesis by A1,Lm21;
end;
