
theorem Th2:
  for I,J be non empty set,
      a be Function of I,J,
      F be multMagma-Family of J
  holds trans_prod(F,a) is multiplicative
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be multMagma-Family of J;
    reconsider f = trans_prod(F,a) as Function of product F, product(F*a);
    for x,y be Element of product F holds f.(x * y) = f.x * f.y
    proof
      let x,y be Element of product F;
      A1: f.(x * y) = (x * y) * a by Def2;
      A2: f.x = x * a by Def2;
      x * a in product(F*a) by Th1; then
      reconsider xa = x * a as Element of product(F*a);
      y * a in product(F*a) by Th1; then
      reconsider ya = y * a as Element of product(F*a);
      A3: dom(xa * ya) = I by GROUP_19:3;
      A4: dom a = I by FUNCT_2:def 1;
      dom(x * y) = J by GROUP_19:3; then
      A5: rng a c= dom(x * y);
      for i be object st i in I holds (xa * ya).i = ((x * y) * a).i
      proof
        let i be object;
        assume i in I; then
        reconsider i as Element of I;
        reconsider Fai = (F*a).i as multMagma;
        reconsider j = a.i as Element of J;
        reconsider Fj = F.j as multMagma;
        xa in product(F*a); then
        xa.i in (F*a).i by GROUP_19:5; then
        reconsider xai = xa.i as Element of Fai;
        ya in product(F*a); then
        ya.i in (F*a).i by GROUP_19:5; then
        reconsider yai = ya.i as Element of Fai;
        x in product F; then
        x.j in F.j by GROUP_19:5; then
        reconsider xj = x.j as Element of Fj;
        y in product F; then
        y.j in F.j by GROUP_19:5; then
        reconsider yj = y.j as Element of Fj;
        A6: xa.i = x.j by A4,FUNCT_1:13;
        A7: ya.i = y.j by A4,FUNCT_1:13;
        ((x * y) * a).i = (x * y).j by A4,FUNCT_1:13
                       .= xj * yj by GROUP_7:1
                       .= xai * yai by A4,A6,A7,FUNCT_1:13
                       .= (xa * ya).i by GROUP_7:1;
        hence thesis;
      end; then
      xa * ya = (x * y) * a by A3,A4,A5,RELAT_1:27;
      hence thesis by A1,A2,Def2;
    end;
    hence thesis by GROUP_6:def 6;
  end;
