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 Th6:
  h1 = g1 implies h1" = g1"
proof
  reconsider g9 = h1" as Element of G by Th2;
A1: h1 * h1" = 1_H1 by GROUP_1:def 5;
  assume h1 = g1;
  then g1 * g9 = 1_H1 by A1,Th3
    .= 1_G by Th4;
  hence thesis by GROUP_1:12;
end;
