
theorem Th10:
  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 bijective
  holds x in sum F iff x*a in sum(F*a) & dom x = J
  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 bijective;
    hereby
      assume
      A2: x in sum F; then
      x in product F by GROUP_2:40;
      hence x*a in sum(F*a) & dom x = J by A1,A2,Th9,GROUP_19:3;
    end;
    assume
    A3: x*a in sum(F*a) & dom x = J;
    A4: rng a = J & a is one-to-one by A1,FUNCT_2:def 3; then
    reconsider b = a" as Function of J,I by FUNCT_2:25;
    A5: a * b = id J & b * a = id I by A4,FUNCT_2:29;
    A6: dom F = J by PARTFUN1:def 2;
    (x*a)*b in sum((F*a)*b) by A3,A1,Th9; then
    x*(a*b) in sum((F*a)*b) by RELAT_1:36; then
    x*(id J) in sum(F*(id J)) by A5,RELAT_1:36; then
    x in sum(F*(id J)) by A3,RELAT_1:52;
    hence x in sum F by A6,RELAT_1:52;
  end;
