
theorem Th1:
  for I,J be non empty set,
      a be Function of I,J,
      F be multMagma-Family of J,
      x be Element of product F
  holds
    x*a in product(F*a)
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be multMagma-Family of J,
        x be Element of product F;
    dom x = J by GROUP_19:3; then
    rng a c= dom x; then
    dom(x*a) = dom a by RELAT_1:27; then
    dom(x*a) = I by PARTFUN1:def 2; then
    reconsider y = x*a as ManySortedSet of I by PARTFUN1:def 2;
    reconsider z = Carrier(F*a) as ManySortedSet of I;
    A1: dom y = I by PARTFUN1:def 2;
    A2: dom z = I by PARTFUN1:def 2;
    A3: dom a = I by PARTFUN1:def 2;
    for i be object st i in I holds y.i in z.i
    proof
      let i be object;
      assume i in I; then
      reconsider i as Element of I;
      reconsider j = a.i as Element of J;
      A4: z.i = [#]((F*a).i) by PENCIL_3:7
             .= [#](F.j) by A3,FUNCT_1:13;
      x in product F; then
      x.j in F.j by GROUP_19:5;
      hence thesis by A3,A4,FUNCT_1:13;
    end; then
    y in product z by A1,A2,CARD_3:def 5;
    hence thesis by GROUP_7:def 2;
  end;
