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

theorem Th39:
  for D be non empty set, F be PartFunc of D,REAL holds F"(
  right_closed_halfline(0)) = max-(F)"{0}
proof
  set li = right_closed_halfline(0);
  let D be non empty set, F be PartFunc of D,REAL;
A1: dom max-(F) = dom F by Def11;
A2: li = {s : 0<=s} by XXREAL_1:232;
  thus F" li c= max-(F)"{0}
  proof
    let x be object;
    assume
A3: x in F" li;
    then reconsider d=x as Element of D;
    F.d in li by A3,FUNCT_1:def 7;
    then ex s st s=F.d & 0<=s by A2;
    then
A4: max(-F.d,0) = 0 by XXREAL_0:def 10;
A5: d in dom F by A3,FUNCT_1:def 7;
    then (max-(F)).d = max-(F.d) by A1,Def11
      .= max(-F.d,0);
    then (max-(F)).d in {0} by A4,TARSKI:def 1;
    hence thesis by A1,A5,FUNCT_1:def 7;
  end;
  let x be object;
  assume
A6: x in (max- F)"{0};
  then reconsider d=x as Element of D;
  (max- F).d in {0} by A6,FUNCT_1:def 7;
  then
A7: (max- F).d = 0 by TARSKI:def 1;
A8: d in dom F by A1,A6,FUNCT_1:def 7;
  then (max- F).d = max-(F.d) by A1,Def11
    .= max(-F.d,0);
  then -F.d <= -0 by A7,XXREAL_0:def 10;
  then 0<=F.d by XREAL_1:24;
  then F.d in li by A2;
  hence thesis by A8,FUNCT_1:def 7;
end;
