reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;

theorem Th9:
  R_EAL F is additive
proof
  now
    let n,m be Nat;
    assume n <> m;
    let x be set;
    assume x in dom((R_EAL F).n) /\ dom((R_EAL F).m);
    (R_EAL F).n = R_EAL(F.n);
    hence ((R_EAL F).n).x <> +infty or ((R_EAL F).m).x <> -infty;
  end;
  hence thesis by MESFUNC9:def 5;
end;
