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
  G is commutative Group implies Image g is commutative
proof
  assume
A1: G is commutative Group;
  let h1,h2 be Element of Image g;
  reconsider c = h1, d = h2 as Element of H by GROUP_2:42;
  h1 in Image g;
  then consider a such that
A2: h1 = g.a by Th45;
  h2 in Image g;
  then consider b such that
A3: h2 = g.b by Th45;
  thus h1 * h2 = c * d by GROUP_2:43
    .= g.(a * b) by A2,A3,Def6
    .= g.(b * a) by A1,Lm2
    .= d * c by A2,A3,Def6
    .= h2 * h1 by GROUP_2:43;
end;
