reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

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;
