
theorem Th11:
  for G,H be Group,
      H0 be Subgroup of H,
      f be Homomorphism of G,H
  st rng f c= [#]H0
  holds f is Homomorphism of G,H0
  proof
    let G,H be Group,
        H0 be Subgroup of H,
        f be Homomorphism of G,H;
    assume rng f c= [#]H0; then
    reconsider g = f as Function of G,H0 by FUNCT_2:6;
    for a,b be Element of G holds g.(a * b) = g.a * g.b
    proof
      let a,b be Element of G;
      g.(a * b) = f.a * f.b by GROUP_6:def 6
               .= g.a * g.b by GROUP_2:43;
      hence thesis;
    end;
    hence thesis by GROUP_6:def 6;
  end;
