
theorem Th3:
  for I,J be non empty set,
      a be Function of I,J,
      F be multMagma-Family of J,
      y be Element of product(F*a)
  st a is bijective
  holds y * a" in product F
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be multMagma-Family of J,
        y be Element of product(F*a);
    assume
    A1: a is bijective; then
    A2: dom a = I & rng a = J by GROUP_6:61; then
    A3: dom(a") = J & rng(a") = I by A1,FUNCT_1:33;
    set x = y * a";
    dom y = I by GROUP_19:3; then
    A4: dom x = J by A3,RELAT_1:27;
    A5: dom(Carrier F) = J by PARTFUN1:def 2;
    for j be object st j in J holds x.j in (Carrier F).j
    proof
      let j be object;
      assume j in J; then
      reconsider j as Element of J;
      consider i be object such that
      A6: i in I & j = a.i by A2,FUNCT_1:def 3;
      reconsider i as Element of I by A6;
      i = (a").j by A1,A2,A6,FUNCT_1:32; then
      A7: x.j = y.i by A3,FUNCT_1:13;
      A8: (Carrier F).j = [#](F.j) by PENCIL_3:7;
      y in product(F*a); then
      y.i in (F*a).i by GROUP_19:5;
      hence thesis by A2,A6,A7,A8,FUNCT_1:13;
    end; then
    x in product(Carrier F) by A4,A5,CARD_3:def 5;
    hence thesis by GROUP_7:def 2;
  end;
