
theorem Th19:
  for G be Group,
  q be set,
  F be associative Group-like multMagma-Family of {q},
  f being Function of G,product F st F = q .--> G &
  for x being Element of G holds f . x = q .--> x holds
  f is Homomorphism of G,(product F)
  proof
    let G be Group,
    q be set,
    F be associative Group-like multMagma-Family of {q},
    f be Function of G, product F;
    assume A1:F = q .--> G;
    assume A2:for x being Element of G holds f . x = q .--> x;
    A3: the carrier of product F = product (Carrier F) by GROUP_7:def 2;
    now
      let a, b be Element of G;
      A4: (f . a) = q .--> a by A2;
      A5: (f . b) = q .--> b by A2;
      reconsider fa=f.a, fb=f.b as Element of product F;
      set ga = q .--> a;
      set gb = q .--> b;
      consider gab be Function such that
      A6: fa*fb = gab & dom gab = dom (Carrier F) &
      for y be object st y in dom (Carrier F)
      holds gab.y in (Carrier F).y by CARD_3:def 5,A3;
      A7: for z being object st z in dom gab holds gab . z = a*b
      proof
        let z be object;
        assume A8:z in dom gab;
        A9: G = F.z by A1,FUNCOP_1:7,A8,A6;
        A10: ga.z = a by FUNCOP_1:7,A8,A6;
        gb.z = b by FUNCOP_1:7,A8,A6;
        hence gab . z = a*b by A4,A5,A6,A8,A9,A10,GROUP_7:1;
      end;
      gab = (dom gab) --> a*b by A7,FUNCOP_1:11
      .= q .--> (a*b) by A6,PARTFUN1:def 2
      .= f . (a * b) by A2;
      hence f . (a * b) = (f . a) * (f . b) by A6;
    end;
    hence thesis by GROUP_6:def 6;
  end;
