
theorem Th7:
  for G,H being AddGroup, h being Homomorphism of G,H st h is bijective holds
  h" is Homomorphism of H,G
proof
  let G,H be AddGroup, h be Homomorphism of G,H;
  assume
A1: h is bijective;
  then
A2: h is one-to-one & rng h = the carrier of H by FUNCT_2:def 3;
  then reconsider h1 = h" as Function of H,G by FUNCT_2:25;
  now
    let a,b be Element of H;
    set a1 = h1.a, b1 = h1.b;
    h.a1 = a & h.b1 = b by A2,FUNCT_1:32;
    hence h1.(a + b) = h1.(h.(a1 + b1)) by VECTSP_1:def 20
      .= h1.a + h1.b by A1,FUNCT_2:26;
  end;
  hence thesis by VECTSP_1:def 20;
end;
