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 Th15:
for C being non empty set, f being PartFunc of C,ExtREAL, x being Element of C
  st 0. < max+(f).x holds max-(f).x = 0.
proof
  let C be non empty set;
  let f be PartFunc of C,ExtREAL;
  let x be Element of C;
A1: dom max+f = dom f by Def2;
  per cases;
  suppose
A2: x in dom f;
    assume
A3: 0. < max+(f).x;
A4: x in dom(max+(f)) by A2,Def2;
A5: x in dom(max-(f)) by A2,Def3;
 max+(f).x = max(f.x,0.) by A4,Def2;
then  not (f.x <= 0. & 0. <= 0.) by A3,XXREAL_0:28;
then  max(-(f.x),0.) = 0. by XXREAL_0:def 10;
    hence thesis by A5,Def3;
  end;
  suppose
 not x in dom f;
    hence thesis by A1,FUNCT_1:def 2;
  end;
end;
