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

theorem Th64:
  for A being non empty set, D being non empty a_partition of A,
    f being finite-support Function of A, REAL
  holds
    support (D eqSumOf f) c= (proj D).:support 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;
  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 A1: eqSumOf F = D eqSumOf f by Def15;
  A2: proj PFP = proj E by Th48, Th51;
  support (eqSumOf F) c= (proj PFP).:support F by Th63;
  hence thesis by A2,A1;
end;
