
theorem LM204G:
  for G, F being Group, H be Subgroup of G, f be Homomorphism of G, F
  holds f| (the carrier of H) is Homomorphism of H, F
  proof
    let G, F be Group, H be Subgroup of G, f be Homomorphism of G, F;
    (the carrier of H) c= the carrier of G by GROUP_2:def 5;
    then reconsider g = f| (the carrier of H)
    as Function of (the carrier of H),(the carrier of F) by FUNCT_2:32;
    for a, b being Element of H holds g.(a * b) = g.a * g.b
    proof
      let a, b be Element of H;
      a in G & b in G by STRUCT_0:def 5, GROUP_2:40;
      then reconsider a0 = a,b0 = b as Element of G;
      A4: f.a0 = g.a by FUNCT_1:49;
      A5: f.b0 = g.b by FUNCT_1:49;
      a*b = a0*b0 by GROUP_2:43;
      hence g.(a * b) = f.(a0*b0) by FUNCT_1:49
      .= (g.a) * (g.b) by GROUP_6:def 6, A4, A5;
    end;
    hence thesis by GROUP_6:def 6;
  end;
