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

theorem Th38:
  for D be non empty set, F be PartFunc of D,REAL, r st 0<
  r holds F"{-r} = max-(F)"{r}
proof
  let D be non empty set, F be PartFunc of D,REAL, r;
A1: dom max-(F) = dom F by Def11;
  assume
A2: 0<r;
  thus F"{-r} c= max-(F)"{r}
  proof
    let x be object;
    assume
A3: x in F"{-r};
    then reconsider d=x as Element of D;
    F.d in {-r} by A3,FUNCT_1:def 7;
    then
A4: F.d = -r by TARSKI:def 1;
A5: d in dom F by A3,FUNCT_1:def 7;
    then (max-(F)).d = max-(F.d) by A1,Def11
      .= r by A2,A4,XXREAL_0:def 10;
    then (max-(F)).d in {r} by TARSKI:def 1;
    hence thesis by A1,A5,FUNCT_1:def 7;
  end;
  let x be object;
  assume
A6: x in (max- F)"{r};
  then reconsider d=x as Element of D;
  (max- F).d in {r} by A6,FUNCT_1:def 7;
  then
A7: (max- F).d = r 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 = r by A2,A7,XXREAL_0:16;
  then F.d in {-r} by TARSKI:def 1;
  hence thesis by A8,FUNCT_1:def 7;
end;
