reserve X for non empty set;
reserve e for set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for Function of RAT,S;
reserve p,q for Rational;
reserve r for Real;
reserve n,m for Nat;
reserve A,B for Element of S;

theorem
  for C being non empty set, f being PartFunc of C,ExtREAL holds
  max-(f) = max+(-f)
proof
  let C be non empty set;
  let f be PartFunc of C,ExtREAL;
A1: dom(max-(f)) = dom f by Def3
    .= dom (-f) by MESFUNC1:def 7;
then A2: dom(max-(f)) = dom(max+(-f)) by Def2;
   for
 x being Element of C st x in dom(max-(f)) holds max-(f).x = max+(-f).x
  proof
    let x be Element of C;
    assume
A3: x in dom (max-(f));
    then  max-
(f).x = max(-(f.x),0.) & -(f.x) = (-f).x by A1,Def3,MESFUNC1:def 7;
    hence thesis by A2,A3,Def2;
  end;
  hence thesis by A2,PARTFUN1:5;
end;
