
theorem VOR:
  for f be real-valued Function, x be object holds
  f.x = (delneg f).x or f.x = -(delpos f).x
  proof
    let f be real-valued Function, x be object;
A2: dom (delneg f) = dom f & dom (delpos f) = dom f by DMN;
    per cases;
    suppose not x in dom f;
      then f.x = {} & (delneg f).x = {} & (delpos f).x = {}
      by A2,FUNCT_1:def 2;
      hence thesis;
    end;
    suppose
    A0: x in dom f;
    A3: dom (abs f) = dom f by VALUED_1:def 11; then
    (dom f)/\dom (abs f) = dom f & (dom (abs f))/\ dom f = dom f; then
    A4: dom (f + abs f) = dom f & dom ((abs f) - f)= dom f
      by VALUED_1:def 1,VALUED_1:12;
    per cases;
    suppose
      B1: f.x >= 0;
      (abs f).x = |.f.x.| by A0,A3,VALUED_1:def 11
      .= f.x by B1,ABSVALUE:def 1; then
      f.x = (1/2)*((f).x + (abs f).x)
      .= (1/2)*(f + (abs f)).x by A0,A4,VALUED_1:def 1
      .= ((1/2)(#)(f + (abs f))).x by VALUED_1:def 5,A0,A2;
      hence thesis;
    end;
    suppose
      B1: f.x < 0;
      (abs f).x = |.f.x.| by A0,A3,VALUED_1:def 11
      .= -f.x by B1,ABSVALUE:def 1; then
      f.x = -(1/2)*((abs f).x - f.x)
      .= -(1/2)*((abs f) - f).x by A0,A4,VALUED_1:13
      .= (-1)*((1/2)*((abs f) - f).x)
      .= (-1)*(((1/2)(#)((abs f) -f)).x) by A0,VALUED_1:def 5,A2;
      hence thesis;
    end;
    end;
  end;
