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 f by A1;
  then
A3: f"{+infty} = {} by FUNCT_1:72;
  not -infty in rng g by A2;
  then g"{-infty} = {} by FUNCT_1:72;
  then g"{+infty} /\ f"{+infty} \/ g"{-infty} /\ f"{-infty} = {} by A3;
  then dom(f-g) = (dom f /\ dom g)\{} by MESFUNC1:def 4;
  hence thesis;
end;
