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

theorem
  for A being Preorder, f being finite-support Function of A, REAL
    st f is nonpositive-yielding
  holds
    eqSumOf f is nonpositive-yielding
proof
  let A be Preorder, f be finite-support Function of A, REAL;
  assume A1: f is nonpositive-yielding;
  reconsider D = the carrier of QuotientOrder(A)
    as a_partition of the carrier of A by Th47;
  (D eqSumOf f) is nonpositive-yielding by A1, Th58;
  hence thesis by Def15;
end;
