
theorem Th36:
  for I be non empty set,
      J be non-empty disjoint_valued ManySortedSet of I,
      F be Group-Family of I,J
  holds sum2dsum F = (prod2dprod F) | 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;
    dsum2sum F is bijective; then
    rng((dprod2prod F) | (dsum F)) = [#] sum(Union F) by FUNCT_2:def 3;
    hence thesis by Th35;
  end;
