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

theorem Th67:
  for A being Preorder,
    f being finite-support Function of A, REAL
  st
    f is nonpositive-yielding
  holds
    (proj A).:support f = support eqSumOf f
proof
  let A be Preorder;
  let f be finite-support Function of the carrier of A, REAL;
  assume A1: f is nonpositive-yielding;
  reconsider mf = -f as finite-support Function of the carrier of A, REAL;
  rng f c= COMPLEX by NUMBERS:11;
  then reconsider fc = f as Function of dom f, COMPLEX by FUNCT_2:2;
  set esof = eqSumOf f;
  rng esof c= COMPLEX by NUMBERS:11;
  then reconsider esofc = esof as
    Function of dom esof, COMPLEX by FUNCT_2:2;
  thus (proj A).:support f = (proj A).:support fc
    .= (proj A).:support mf by Th10
    .= support eqSumOf mf by Th65, A1
    .= support -esofc by Th55
    .= support eqSumOf f by Th10;
end;
