
theorem Th5:
  for I,J be non empty set,
      F be multMagma-Family of J,
      a be Function of I,J st a is bijective
  holds dom trans_prod(F,a) = [#] product(F)
      & rng trans_prod(F,a) = [#] product(F*a)
  proof
    let I,J be non empty set,
        F be multMagma-Family of J,
        a be Function of I,J;
    assume
    A1: a is bijective;
    set f = trans_prod(F,a);
    for y be object st y in [#] product(F*a) holds
    ex x be object st x in [#] product(F) & y = f.x
    proof
      let y be object;
      assume y in [#] product(F*a); then
      reconsider y as Element of product(F*a);
      set x = y * a";
      x in product F by A1,Th3; then
      reconsider x as Element of product F;
      dom x = J & dom y = I by GROUP_19:3; then
      y = x * a by A1,Th4; then
      y = f.x by Def2;
      hence thesis;
    end;
    hence thesis by FUNCT_2:10,def 1;
  end;
