reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;

theorem Th50:
  for X being bounded_below non empty ext-real-membered set st X
  <> {+infty} holds ex x being Element of REAL st x in X
proof
  let X be bounded_below non empty ext-real-membered set;
  assume X <> {+infty};
  then consider x being object such that
A1: x in X and
A2: x <> +infty by ZFMISC_1:35;
  reconsider x as ExtReal by A1;
  consider r being Real such that
A3: r is LowerBound of X by Def9;
A4: r in REAL by XREAL_0:def 1;
  r <= x by A3,A1,Def2;
  then x in REAL by A4,A2,XXREAL_0:10;
  hence thesis by A1;
end;
