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

theorem Th15:
  for X being non empty real-membered set st X is bounded_below
  holds inf X in REAL
proof
  let X be non empty real-membered set;
  given r being Real such that
A1: r is LowerBound of X;
  consider s being Real such that
A2: s in X by MEMBERED:9;
A3: inf X <= s by A2,Th3;
A4: r in REAL by XREAL_0:def 1;
A5: s in REAL by XREAL_0:def 1;
  r <= inf X by A1,Def4;
  hence thesis by A4,A5,A3,XXREAL_0:45;
end;
