theorem
  for C being non empty set, f being PartFunc of C,ExtREAL,
  x being Element of C st x in dom f & max-(f).x = 0. holds max+(f).x = f.x
proof
  let C be non empty set;
  let f be PartFunc of C,ExtREAL;
  let x be Element of C;
  assume that
A1: x in dom f and
A2: max-(f).x = 0.;
A3: x in dom(max+(f)) by A1,Def2;
A4: x in dom(max-(f)) by A1,Def3;
A5: max+(f).x = max(f.x,0.) by A3,Def2;
 max-(f).x = max(-(f.x),0.) by A4,Def3;
then  -0. <= -(-(f.x)) by A2,XXREAL_0:def 10;
  hence thesis by A5,XXREAL_0:def 10;
end;
