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 st (for d be Element
  of D st d in dom F holds F.d<=0) holds max- F = -F
proof
  let D be non empty set, F be PartFunc of D,REAL;
A1: dom(max- F) = dom F by Def11;
  assume
A2: for d be Element of D st d in dom F holds F.d<=0;
A3: now
    let d be Element of D;
    assume
A4: d in dom F;
    then
A5: F.d<=0 by A2;
    thus (max- F).d = max-(F.d) by A1,A4,Def11
      .= -F.d by A5,XXREAL_0:def 10
      .= (-F).d by VALUED_1:8;
  end;
  dom F = dom(-F) by VALUED_1:8;
  hence thesis by A1,A3,PARTFUN1:5;
end;
