
theorem Th9:
  for I,J be non empty set,
      a be Function of I,J,
      F be Group-Family of J,
      x be Function
  st a is one-to-one
  holds x in sum F implies x*a in sum(F*a)
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be Group-Family of J,
        x be Function;
    assume
    A1: a is one-to-one;
    assume
    A2: x in sum F; then
    x in product F by GROUP_2:40; then
    reconsider x as Element of product F;
    reconsider Fa = F*a as Group-Family of I;
    x*a in product(F*a) by Th1; then
    reconsider xa = x*a as Element of product(F*a);
    A3: dom a = I by FUNCT_2:def 1;
    A4: a .: support(xa,Fa) c= support(x,F) by Th7;
    a .: support(xa,Fa), support(xa,Fa) are_equipotent by A1,A3,CARD_1:33; then
    support(xa,Fa) is finite by A2,A4,CARD_1:38;
    hence thesis by GROUP_19:8;
  end;
