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

theorem Th49:
  for X being bounded_above 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_above 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 UpperBound of X by Def10;
A4: r in REAL by XREAL_0:def 1;
  x <= r by A3,A1,Def1;
  then x in REAL by A4,A2,XXREAL_0:13;
  hence thesis by A1;
end;
