
theorem Th6:
  for I,J be non empty set,
      a be Function of I,J,
      F be multMagma-Family of J
  st a is bijective
  holds trans_prod(F,a) is bijective
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be multMagma-Family of J;
    assume
    A1: a is bijective;
    reconsider f = trans_prod(F,a) as Function of product F, product(F*a);
    A2: dom f = [#](product F) & rng f = [#] product(F*a) by A1,Th5;
    for x,y be object st x in dom f & y in dom f & f.x = f.y holds x = y
    proof
      let x,y be object;
      assume that
      A3: x in dom f and
      A4: y in dom f and
      A5: f.x = f.y;
      reconsider x as Element of product F by A3;
      reconsider y as Element of product F by A4;
      A6: dom x = J & dom y = J by GROUP_19:3;
      A7: rng a = J by A1,FUNCT_2:def 3;
      f.x = x * a by Def2; then
      x * a = y * a by A5,Def2;
      hence thesis by A6,A7,FUNCT_1:86;
    end; then
    A8: f is one-to-one;
    f is onto by A2,FUNCT_2:def 3;
    hence thesis by A8;
  end;
