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

theorem Th54:
  for A being set, D being a_partition of A,
    f being finite-support Function of A, REAL
  holds
    (D eqSumOf -f) = -(D eqSumOf f)
proof
  let A be set;
  let D be a_partition of A;
  let f being finite-support Function of A, REAL;
  dom (D eqSumOf -f) = D by FUNCT_2:def 1;
  then A1: dom (D eqSumOf -f) = dom -(D eqSumOf f) by FUNCT_2:def 1;
  for X being object st X in dom (D eqSumOf -f) holds
    (D eqSumOf -f).X = (-(D eqSumOf f)).X
  proof
    let X be object;
    assume A2: X in dom (D eqSumOf -f);
    then reconsider Y = X as Element of D;
    set s = canFS(eqSupport(f,Y));
    set t = canFS(eqSupport(-f,Y));
    A3: dom f = A by FUNCT_2:def 1;
    A4: rng s = eqSupport(f,Y) by FUNCT_2:def 3;
    A5: rng t = eqSupport(-f,Y) by FUNCT_2:def 3;
    A6: dom s = Seg len s by FINSEQ_1:def 3
      .= Seg len t by Th52
      .= dom t by FINSEQ_1:def 3;
    A7: rng s c= dom f & rng t c= dom f by A3, A4, A5;
    s, t are_fiberwise_equipotent by Th52;
    then A8: f*s, f*t are_fiberwise_equipotent by A7, A6, CLASSES1:83;
    A9: rng (f*s) c= REAL & rng (f*t) c= REAL;
    A11: dom ((-f)*t) = dom -(f*t)
    proof
      for x being object holds x in dom ((-f)*t) iff x in dom -(f*t)
      proof
        rng f c= COMPLEX by NUMBERS:11;
        then reconsider fc = f as Function of dom f, COMPLEX by FUNCT_2:2;
        let x be object;
        hereby
          assume x in dom ((-f)*t);
          then x in dom t & t.x in dom (-fc) by FUNCT_1:11;
          then x in dom (fc*t) by FUNCT_1:11;
          hence x in dom -(f*t) by CFUNCT_1:5;
        end;
        assume x in dom -(f*t);
        then x in dom (fc*t) by CFUNCT_1:5;
        then x in dom t & t.x in dom fc by FUNCT_1:11;
        then x in dom t & t.x in dom (-fc) by CFUNCT_1:5;
        hence x in dom ((-f)*t) by FUNCT_1:11;
      end;
      hence thesis by TARSKI:2;
    end;
    for x being object st x in dom ((-f)*t) holds ((-f)*t).x = (-(f*t)).x
    proof
      let x be object;
      set domft = dom (f*t);
      rng (f*t) c= COMPLEX by NUMBERS:11;
      then reconsider ftc = f*t as Function of domft, COMPLEX by FUNCT_2:2;
      assume A12: x in dom ((-f)*t);
      then a13: x in dom -ftc by A11; then
      reconsider domft as non empty set;
      dom f is non empty
      proof
        A is non empty by A2;
        hence thesis;
      end;
      then reconsider domf = dom f as non empty set;
      reconsider tc = t.x as Element of domf by a13, FUNCT_1:11;
      reconsider c = x as Element of domft by a13;
      reconsider F = f as Function of domf, REAL by FUNCT_2:def 1;
      reconsider FT = f*t as Function of domft, REAL by A9, FUNCT_2:2;
      thus ((-f)*t).x = (-f).(t.x) by A12, FUNCT_1:12
        .= - (F.tc) by RFUNCT_1:58
        .= - ((FT).c) by FUNCT_1:12
        .= (-(f*t)).x by RFUNCT_1:58;
    end;
    then A14: (-f)*t = -(f*t) by A11, FUNCT_1:2;
    thus (D eqSumOf -f).X = Sum ((-f)*t) by A2, Def14
      .= - Sum (f*t) by A14, RVSUM_1:88
      .= - Sum (f*s) by A8, RFINSEQ:9
      .= -(D eqSumOf f).Y by A2, Def14
      .= (-(D eqSumOf f)).X by A2, RFUNCT_1:58;
  end;
  hence thesis by A1, FUNCT_1:2;
end;
