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

theorem Th58:
  for X being non empty ext-real-membered set st X is
  bounded_below & X <> {+infty} holds inf X in REAL
proof
  let X be non empty ext-real-membered set;
  assume
A1: X is bounded_below;
  then consider r being Real such that
A2: r is LowerBound of X;
  assume X <> {+infty};
  then
A3: ex x being Element of REAL st x in X by A1,Th50;
  inf X is LowerBound of X by Def4;
  then
A4: inf X <> +infty by A3,Def2,XXREAL_0:9;
A5: r in REAL by XREAL_0:def 1;
  r <= inf X by A2,Def4;
  hence thesis by A5,A4,XXREAL_0:10;
end;
