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 Th48:
  for N being strict normal StableSubgroup of G holds Ker nat_hom N = N
proof
  let N be strict normal StableSubgroup of G;
  reconsider N9 = the multMagma of N as strict normal Subgroup of G by Lm6;
A1: nat_hom N = nat_hom N9 & 1_(G./.N) = 1_(G./.N9) by Def20,Lm34;
  the carrier of Ker nat_hom N = {a where a is Element of G: (nat_hom N).a
  = 1_(G./.N)} by Def21
    .= {a where a is Element of G: (nat_hom N9).a = 1_(G./.N9)} by A1
    .= the carrier of Ker nat_hom N9 by GROUP_6:def 9
    .= the carrier of N by GROUP_6:43;
  hence thesis by Lm4;
end;
