reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem
  for D be non empty set, F be PartFunc of D,REAL holds F = max+(F) -
  max-(F) & abs(F) = max+(F) + max-(F) & 2 (#) max+(F) = F + abs(F)
proof
  let D be non empty set, F be PartFunc of D,REAL;
A1: dom F = dom F /\ dom F;
A2: dom max+(F) = dom F by Def10;
A3: dom max-(F) = dom F by Def11;
  dom -max-(F) = dom max-(F) by VALUED_1:def 5;
  then
A4: dom F =dom(max+(F) + -max-(F)) by A2,A3,A1,VALUED_1:def 1;
  now
    let d be Element of D;
    assume
A5: d in dom F;
    hence (max+(F) - max-(F)).d = max+(F).d - max-(F).d by A4,VALUED_1:13
      .= max+(F.d) - max-(F).d by A2,A5,Def10
      .= max+(F.d) - max-(F.d) by A3,A5,Def11
      .= F.d by Th1;
  end;
  hence F = max+(F) - max-(F) by A4,PARTFUN1:5;
A6: dom abs(F) = dom F by VALUED_1:def 11;
  then
A7: dom abs(F) = dom(max+(F) + max-(F)) by A2,A3,A1,VALUED_1:def 1;
  now
    let d be Element of D;
    assume
A8: d in dom abs(F);
    hence (max+(F) + max-(F)).d = max+(F).d + max-(F).d by A7,VALUED_1:def 1
      .= max+(F.d) + max-(F).d by A2,A6,A8,Def10
      .= max+(F.d) + max-(F.d) by A3,A6,A8,Def11
      .= |.F.d.| by Th2
      .= (abs(F)).d by VALUED_1:18;
  end;
  hence abs(F) = max+(F) + max-(F) by A7,PARTFUN1:5;
A9: dom(2(#)max+(F)) = dom max+(F) by VALUED_1:def 5;
  then
A10: dom(2(#)max+(F)) = dom(F + abs(F)) by A2,A6,A1,VALUED_1:def 1;
  now
    let d be Element of D;
    assume
A11: d in dom F;
    hence (2(#)max+(F)).d = 2*(max+(F).d) by A2,A9,VALUED_1:def 5
      .=2*max+(F.d) by A2,A11,Def10
      .= F.d + |.F.d.| by Th3
      .= F.d + (abs(F)).d by VALUED_1:18
      .=(F+abs(F)).d by A2,A9,A10,A11,VALUED_1:def 1;
  end;
  hence thesis by A2,A9,A10,PARTFUN1:5;
end;
