
theorem Th13:
  for G be Group,
      I,J be non empty set,
      F be DirectSumComponents of G,J,
      a be Function of I,J st a is bijective
  holds F * a is DirectSumComponents of G,I
  proof
    let G be Group,
        I,J be non empty set,
        F be DirectSumComponents of G,J,
        a be Function of I,J;
    assume
    A1: a is bijective;
    consider h1 be Homomorphism of sum F,G such that
    A2: h1 is bijective by GROUP_19:def 8;
    reconsider h2 = trans_sum(F,a) as Homomorphism of sum F,sum(F*a) by A1,TT;
    A3: h2 is bijective by A1,Th12;
    then reconsider h3 = h2" as Homomorphism of sum(F*a),sum F
      by GROUP_6:62;
    reconsider h = h1 * h3 as Homomorphism of sum(F*a),G;
    h3 is bijective by A3,GROUP_6:63; then
    h is bijective by A2,GROUP_6:64;
    hence thesis by GROUP_19:def 8;
  end;
