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

theorem Th57:
  for X being non empty ext-real-membered set st X is
  bounded_above & X <> {-infty} holds sup X in REAL
proof
  let X be non empty ext-real-membered set;
  assume
A1: X is bounded_above;
  then consider r being Real such that
A2: r is UpperBound of X;
  assume X <> {-infty};
  then
A3: ex x being Element of REAL st x in X by A1,Th49;
  sup X is UpperBound of X by Def3;
  then
A4: not sup X = -infty by A3,Def1,XXREAL_0:12;
A5: r in REAL by XREAL_0:def 1;
  sup X <= r by A2,Def3;
  hence thesis by A5,A4,XXREAL_0:13;
end;
