reserve X,Z for set;
reserve x,y,z for object;
reserve A,B,C for Subset of X;

theorem
  for A being non empty set, D being non empty a_partition of A,
    f being finite-support Function of A, REAL
  st
    f is nonpositive-yielding
  holds
    (proj D).:support f = support (D eqSumOf f)
proof
  let A be non empty set, D be non empty a_partition of A;
  let f be finite-support Function of A, REAL;
  assume A1: f is nonpositive-yielding;
  reconsider PFP = PreorderFromPartition(D) as non empty Preorder;
  reconsider F = f as finite-support Function of PFP, REAL;
  reconsider E = D as a_partition of the carrier of PFP;
  D = the carrier of QuotientOrder(PFP) by Th51;
  then A2: eqSumOf F = D eqSumOf f by Def15;
  A3: proj PFP = proj E by Th48, Th51;
  support (eqSumOf F) = (proj PFP).:support F by A1, Th67;
  hence thesis by A3, A2;
end;
