
theorem
  for eps being ExtReal, X being non empty Subset of ExtREAL st
  0. < eps & sup X is Real holds ex x being ExtReal st
  x in X & sup X - eps < x
proof
  let eps be ExtReal;
  let X be non empty Subset of ExtREAL;
  assume that
A1: 0. < eps and
A2: sup X is Real;
A3: sup X in REAL by A2,XREAL_0:def 1;
  assume not ex x being ExtReal st x in X & sup X - eps < x;
  then sup X - eps is UpperBound of X by XXREAL_2:def 1;
  then
A4: sup X <= sup X - eps by XXREAL_2:def 3;
  per cases by XXREAL_0:4;
  suppose
    eps < +infty;
    then reconsider a = sup X, b = eps as Element of REAL by A1,A3,XXREAL_0:48;
    a <= a - b by A4,SUPINF_2:3;
    hence thesis by A1,XREAL_1:44;
  end;
  suppose
    eps = +infty;
    then sup X - eps = -infty by A3,XXREAL_3:13;
    hence thesis by A3,A4,XXREAL_0:6;
  end;
end;
