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;
