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 f-g is without-infty
proof
  assume that
A1: f is without-infty and
A2: g is without+infty;
  for x be set st x in dom(f-g) holds -infty < (f-g).x
  proof
    let x be set;
    assume
A3: x in dom(f-g);
A4: -infty < f.x by A1;
A5: g.x < +infty by A2;
    (f-g).x = f.x - g.x by A3,MESFUNC1:def 4;
    hence thesis by A4,A5,XXREAL_0:6,XXREAL_3:19;
  end;
  hence thesis by MESFUNC5:10;
end;
