theorem Th48:
  for N being normal Subgroup of G holds Image nat_hom N = G./.N
proof
  let N be normal Subgroup of G;
  now
    let S be Element of G./.N;
    consider a such that
A1: S = a * N and
    S = N * a by Th13;
    (nat_hom N).a = a * N by Def8;
    hence S in Image nat_hom N by A1,Th45;
  end;
  hence thesis by GROUP_2:62;
end;
