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 Th54:
  nat_hom (1).G is bijective
proof
  reconsider H = the multMagma of (1).G as strict normal Subgroup of G by Lm6;
  set g = nat_hom (1).G;
  reconsider G9=G as Group;
A1: the carrier of H = {1_G9} by Def8;
A2: nat_hom (1).G9 is bijective & g is onto by Th53,GROUP_6:65;
  nat_hom (1).G = nat_hom H by Def20
    .= nat_hom (1).G9 by A1,GROUP_2:def 7;
  hence thesis by A2;
end;
