
theorem Th3:
  for x,y be R_eal, e be Real st |.x-y.| < e & not ( x =
  +infty & y = +infty or x = -infty & y = -infty ) holds x <> +infty & x <>
  -infty & y <> +infty & y <>-infty
proof
  let x,y be R_eal, e be Real;
  assume
A1: |.x-y.| < e;
  y-x <= |.x-y.| by Th2;
  then
A2: y-x < e by A1,XXREAL_0:2;
  x-y <= |.x-y.| by EXTREAL1:20;
  then x-y < e by A1,XXREAL_0:2;
  hence thesis by A2,XXREAL_3:54;
end;
