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

theorem Th36:
  for D be non empty set, F be PartFunc of D,REAL holds F"(
  left_closed_halfline(0)) = max+(F)"{0}
proof
  set li = left_closed_halfline(0);
  let D be non empty set, F be PartFunc of D,REAL;
A1: dom max+(F) = dom F by Def10;
A2: li = {s : s<=0} by XXREAL_1:231;
  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 & s<=0 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,Def10
      .= 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,Def10
    .= max(F.d,0);
  then F.d <= 0 by A7,XXREAL_0:def 10;
  then F.d in li by A2;
  hence thesis by A8,FUNCT_1:def 7;
end;
