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 Th50:
  Image nat_hom N = G./.N
proof
  reconsider N9 = the multMagma of N as strict normal Subgroup of G by Lm6;
  reconsider H = G./.N as strict StableSubgroup of G./.N by Lm3;
A1: G./.N9 = the multMagma of G./.N by Lm34;
  the carrier of Image nat_hom N = nat_hom N .: (the carrier of G) by Def22
    .= nat_hom N9 .: (the carrier of G) by Def20
    .= the carrier of Image nat_hom N9 by GROUP_6:def 10
    .= the carrier of H by A1,GROUP_6:48;
  hence thesis by Lm4;
end;
