
theorem Th32:
  for I be non empty set,
      J be non-empty disjoint_valued ManySortedSet of I,
      F be Group-Family of I,J,
      x,y be Function st x in dsum F & x in dsum F
  holds (dprod2prod F).x in sum(Union F)
  proof
    let I be non empty set,
        J be non-empty disjoint_valued ManySortedSet of I,
        F be Group-Family of I,J,
        x,y be Function;
    assume
    A1: x in dsum F; then
    A2: x in dprod(F) by GROUP_2:40; then
    reconsider y = (dprod2prod F).x as Element of product(Union F)
    by FUNCT_2:5;
    deffunc P(object) = support(y | (J.In($1,I)), F.In($1,I));
    set sry = supp_restr(y,F);
    reconsider sry1 = sry | support(x,sum_bundle F) as non-empty
      disjoint_valued ManySortedSet of support(x,sum_bundle F) by A2,Th29;
    for i be object st i in dom sry1 holds sry1.i is finite
    proof
      let i be object;
      assume
      A3: i in dom sry1; then
      i in support(x,sum_bundle F); then
      reconsider i as Element of I;
      A4: y | (J.i) = x.i by A2,Def10;
      x.i in (sum_bundle F).i by A1,GROUP_2:40,GROUP_19:5; then
      A5: x.i in sum(F.i) by Def7; then
      x.i in product(F.i) by GROUP_2:40; then
      reconsider zi = x.i as Element of product (F.i);
      sry.i = support(y | (J.i), F.i) by Def12;
      hence thesis by A3,A4,A5,FUNCT_1:49;
    end; then
    Union sry1 is finite by A1,CARD_2:88; then
    support(y,Union F) is finite by A2,Th29;
    hence thesis by GROUP_19:8;
  end;
