reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;

theorem
  for x being set st x in dom f holds (max+f.x = f.x or max+f.x = 0) &
  (max-f.x = -(f.x) or max-f.x = 0)
proof
  let x be set;
  assume
A1: x in dom f;
  then max+(R_EAL f).x = (R_EAL f).x or max+(R_EAL f).x = 0. by MESFUNC2:18;
  hence max+f.x = f.x or max+f.x = 0 by Th30;
A2: max+(R_EAL f) = max+f & max-(R_EAL f) = max-f by Th30;
  max-(R_EAL f).x = -((R_EAL f).x) or max-(R_EAL f).x = 0. by A1,MESFUNC2:18;
  hence thesis by A2,SUPINF_2:2;
end;
