
theorem
  for eps being ExtReal, X being non empty Subset of ExtREAL st
  0. < eps & inf X is Real holds
  ex x being ExtReal st x in X & x < inf X + eps
proof
  let eps be ExtReal;
  let X be non empty Subset of ExtREAL;
  assume that
A1: 0. < eps and
A2: inf X is Real;
A3: inf X in REAL by A2,XREAL_0:def 1;
  assume not ex x being ExtReal st x in X & x < inf X + eps;
  then inf X + eps is LowerBound of X by XXREAL_2:def 2;
  then
A4: inf X + eps <= inf X by XXREAL_2:def 4;
  per cases by XXREAL_0:4;
  suppose
    eps < +infty;
    then reconsider a = inf X, b = eps as Element of REAL by A1,A3,XXREAL_0:48;
    inf X + eps = a + b by SUPINF_2:1;
    then b <= a - a by A4,XREAL_1:19;
    hence thesis by A1;
  end;
  suppose
    eps = +infty;
    then inf X + eps = +infty by A3,XXREAL_3:def 2;
    hence thesis by A3,A4,XXREAL_0:4;
  end;
end;
