reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;

theorem
  A c= ].r,s.[ implies A is bounded_below
  proof
    assume
A1: A c= ].r,s.[;
    ].r,s.[ c= [.r,s.[ by XXREAL_1:22;
    then A c= [.r,s.[ by A1;
    hence thesis by XXREAL_2:44;
  end;
