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

theorem Th65:
  for A being Preorder,
    f being finite-support Function of A, REAL
  st
    f is nonnegative-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 nonnegative-yielding;
  for X being object holds
    X in (proj A).:support f implies X in support eqSumOf f
  proof
    let X be object;
    assume A2: X in (proj A).:support f;
    then consider x being object such that
      A3: x in dom proj A and
      A4: x in support f and
      A5: X = (proj A).x by FUNCT_1:def 6;
    A6: X in the carrier of QuotientOrder(A) by A2, FUNCT_2:36, TARSKI:def 3;
    reconsider Y = X as Element of the carrier of QuotientOrder(A) by A6;
    set s = canFS(eqSupport(f,Y));
    A7: rng s = eqSupport(f, Y) by FUNCT_2:def 3;
    s is FinSequence of the carrier of A by FINSEQ_2:24;
    then reconsider fs = f*s as FinSequence of REAL by FINSEQ_2:32;
    A8: ex k being Nat st k in dom fs & fs.k <> 0
    proof
      reconsider y = x as Element of A by A3;
      X = Class(EqRelOf A, y) by A5, Def8;
      then y in Y by A3, EQREL_1:20;
      then y in eqSupport(f, Y) by A4, XBOOLE_0:def 4;
      then consider i being object such that
        A9: i in dom s and
        A10: s.i = y by A7, FUNCT_1:def 3;
      reconsider i as Nat by A9;
      take i;
      thus i in dom fs by A4, A10, A9, FUNCT_1:11;
      f.y <> 0 by A4, PRE_POLY:def 7;
      hence fs.i <> 0 by A9, A10, FUNCT_1:13;
    end;
    A11: (eqSumOf f).Y = Sum fs by A6, Def16;
    Sum fs > 0 by A1, A8, Th7;
    hence X in support eqSumOf f by PRE_POLY:def 7, A11;
  end;
  then (proj A).:support f c= support eqSumOf f;
  hence thesis by Th63;
end;
