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 Th15:
  (1).G = (1).H1
proof
A1: 1_H1 = 1_G by Th4;
  (1).H1 is StableSubgroup of G & the carrier of (1).H1 = {1_H1} by Def8,Th11;
  hence thesis by A1,Def8;
end;
