
theorem Th6:
  for G,H be Group,
      I be Subgroup of H,
      f be Homomorphism of G,I
  holds f is Homomorphism of G,H
  proof
    let G,H be Group,
        I be Subgroup of H,
        f be Homomorphism of G,I;
    [#]I c= [#]H by GROUP_2:def 5; then
    reconsider g = f as Function of G,H by FUNCT_2:7;
    now
      let a,b be Element of G;
      thus g.(a*b) = f.a * f.b by GROUP_6:def 6
                  .= g.a * g.b by GROUP_2:43;
    end;
    hence thesis by GROUP_6:def 6;
  end;
