
theorem TT:
  for I,J be non empty set
  for a be Function of I,J
  for F be Group-Family of J st a is bijective holds
  trans_sum(F,a) is Homomorphism of sum F, sum(F*a)
  proof
  let I,J be non empty set;
  let a be Function of I,J;
  let F be Group-Family of J;
  assume
  A1: a is bijective;
    set f = trans_sum(F,a);
    A2: f = trans_prod(F,a) | (sum F) by A1,Def3;
    rng f c= [#] sum(F*a);
    hence thesis by A2,Th11;
  end;
