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;
