A1: ].r,s.[ is bounded_above

proof

take s ; :: according to XXREAL_2:def 10 :: thesis: s is UpperBound of ].r,s.[
let x be ExtReal; :: according to XXREAL_2:def 1 :: thesis: ( not x in ].r,s.[ or x <= s )
thus ( not x in ].r,s.[ or x <= s ) by XXREAL_1:4; :: thesis: verum

end;

].r,s.[ is bounded_below

proof

take r ; :: according to XXREAL_2:def 9 :: thesis: r is LowerBound of ].r,s.[
let x be ExtReal; :: according to XXREAL_2:def 2 :: thesis: ( not x in ].r,s.[ or r <= x )
thus ( not x in ].r,s.[ or r <= x ) by XXREAL_1:4; :: thesis: verum

end;

hence ].r,s.[ is real-bounded by A1; :: thesis: verum