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 Th7:
  for f,g,A st f is real-valued & g is real-valued & f is A-measurable &
  g is A-measurable holds f+g is A-measurable
proof
  let f,g,A;
  assume that
A1: f is real-valued & g is real-valued and
A2: f is A-measurable & g is A-measurable;
 for r be Real holds A /\ less_dom(f+g, r) in S
  proof
    let r be Real;
    reconsider r as Real;
    consider F being Function of RAT,S such that
A3: for p being Rational holds
    F.p = (A /\ less_dom(f, p)) /\ (A /\ less_dom(g, (r-p)))
    by A2,Th6;
    consider G being sequence of S such that
A4: rng F = rng G by Th5,MESFUNC1:5;
 A /\ less_dom(f+g, r) = union (rng G) by A1,A3,A4,Th3;
    hence thesis;
  end;
  hence thesis;
end;
