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;
