
theorem Th12:
  for I,J be non empty set,
      a be Function of I,J,
      F be Group-Family of J
  st a is bijective
  holds trans_sum(F,a) is bijective
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be Group-Family of J;
    assume
    A1: a is bijective;
    reconsider f = trans_prod(F,a) as Homomorphism of product F,product(F*a);
    reconsider g = trans_sum(F,a) as Homomorphism of sum F,sum(F*a) by A1,TT;
    A2: g = f | (sum F) by A1,Def3;
    f is bijective by A1,Th6; then
    A3: g is one-to-one by A2,FUNCT_1:52;
    for y be object st y in [#] sum(F*a) holds y in rng g
    proof
      let y be object;
      assume
      A4: y in [#] sum(F*a); then
      reconsider y as Element of product(F*a) by GROUP_2:42;
      set x = y * a";
      x in product F by A1,Th3; then
      reconsider x as Element of product F;
      A5: dom g = [#] sum F by FUNCT_2:def 1;
      A6: dom x = J & dom y = I by GROUP_19:3; then
      A7: y = x * a by A1,Th4;
      y in sum(F*a) by A4; then
      A8: x in sum F by A1,A6,A7,Th10;
      y = f.x by A7,Def2; then
      y = g.x by A2,A8,FUNCT_1:49;
      hence thesis by A5,A8,FUNCT_1:def 3;
    end; then
    [#] sum(F*a) c= rng g; then
    g is onto by FUNCT_2:def 3,XBOOLE_0:def 10;
    hence thesis by A3;
  end;
