reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem
  f is without+infty & g is without+infty implies dom(f+g)=dom f /\ dom g
proof
  assume that
A1: f is without+infty and
A2: g is without+infty;
  not +infty in rng g by A2;
  then
A3: g"{+infty} = {} by FUNCT_1:72;
  not +infty in rng f by A1;
  then f"{+infty} = {} by FUNCT_1:72;
  then f"{+infty} /\ g"{-infty} \/ f"{-infty} /\ g"{+infty} = {} by A3;
  then dom(f+g) = (dom f /\ dom g)\{} by MESFUNC1:def 3;
  hence thesis;
end;
