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= ].s,r.[ implies A is bounded_above
  proof
    assume
A1: A c= ].s,r.[;
    ].s,r.[ c= ].s,r.] by XXREAL_1:21;
    then A c= ].s,r.] by A1;
    hence thesis by XXREAL_2:43;
  end;
