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 & 0 < max+f.x holds max-f.x = 0
proof
  let x be set;
  assume that
A1: x in dom f and
A2: 0 < max+(f).x;
  0. < (max+(R_EAL(f))).x by A2,Th30;
  then max-(R_EAL(f)).x = 0. by A1,MESFUNC2:15;
  hence thesis by Th30;
end;
