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;

theorem
  for F be Function of RAT,S st (for p holds F.p = (A/\less_dom(f,p)) /\
  (A/\less_dom(g,r-p))) holds A /\ less_dom(f+g,r) = union rng F
proof
  let F be Function of RAT,S;
  assume for p holds F.p = (A /\ less_dom(f,p)) /\ (A /\ less_dom(g,r-p));
  then for p holds F.p = (A /\ less_dom(R_EAL f,p)) /\ (A /\ less_dom(
  R_EAL g,r-p));
  then
A1: A /\ less_dom(R_EAL f + R_EAL g,r) = union rng F by MESFUNC2:3;
  R_EAL f + R_EAL g = R_EAL(f+g) by Th23;
  hence thesis by A1;
end;
