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 ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL ,
  A,B for Element of S,
  r,s for Real;

theorem
  A c= dom f /\ dom g & f is A-measurable & g is A-measurable
  implies max-(f+g) + max+f is A-measurable
proof
  assume that
A1: A c= dom f /\ dom g and
A2: f is A-measurable and
A3: g is A-measurable;
A4: max+ f is A-measurable by A2,Th46;
A5: dom(f+g) = dom f /\ dom g by VALUED_1:def 1;
  f+g is A-measurable by A2,A3,Th26;
  then max-(f+g) is A-measurable by A1,A5,Th47;
  hence thesis by A4,Th26;
end;
