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 ext-real number ; :: 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_1:3; :: 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 ext-real number ; :: 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_1:3; :: thesis: verum

end;

hence for b₁ being Subset of REAL st b₁ = [.r,s.[ holds
b₁ is bounded by A1; :: thesis: verum