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

theorem Th63:
  for A being Preorder,
    f being finite-support Function of A, REAL
  holds
    support eqSumOf f c= (proj A).:support f
proof
  let A be Preorder;
  let f be finite-support Function of the carrier of A, REAL;
  for X being object holds X in support eqSumOf f implies
    X in (proj A).:support f
  proof
    let X be object;
    assume A1: X in support eqSumOf f;
    ex x being object st x in dom proj A & x in support f & X = (proj A).x
    proof
      X in dom eqSumOf f by A1;
      then A2: X in the carrier of QuotientOrder(A);
      A3: dom proj A = the carrier of A by A2, FUNCT_2:def 1;
      reconsider Y = X as Element of QuotientOrder(A) by A2;
      set s = canFS(eqSupport(f,Y));
      A4: rng s c= eqSupport(f, Y) by FINSEQ_1:def 4;
      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;
      (eqSumOf f).Y <> 0 by A1, PRE_POLY:def 7;
      then Sum (fs) <> 0 by A2, Def16;
      then consider i being Nat such that
        A5: i in dom (fs) and
        A6: (fs).i <> 0 by Th6;
      A7: i in dom s & s.i in dom f by A5, FUNCT_1:11;
      then reconsider x = s.i as Element of A;
      take x;
      thus x in dom proj A by A3, A7;
      f.x <> 0 by A6, A7, FUNCT_1:13;
      hence x in support f by PRE_POLY:def 7;
      x in eqSupport(f, Y) by A7, A4, FUNCT_1:3;
      then A8: x in Y by XBOOLE_1:17, TARSKI:def 3;
      X in Class EqRelOf A by A2, Def7;
      then consider y being object such that
        A9: y in the carrier of A and
        A10: X = Class(EqRelOf A, y) by EQREL_1:def 3;
      A11: x in Class(EqRelOf A, y) by A8, A10;
      thus (proj A).x = Class(EqRelOf A, x) by Def8
        .= X by A10, A9, A11, EQREL_1:23;
    end;
    hence thesis by FUNCT_1:def 6;
  end;
  hence thesis;
end;
