set A = right_open_halfline a;
thus right_open_halfline a is bounded_below :: thesis: ( not right_open_halfline a is bounded_above & right_open_halfline a is interval )

proof

take a ; :: according to XXREAL_2:def 9 :: thesis: a is LowerBound of right_open_halfline a
let x be ExtReal; :: according to XXREAL_2:def 2 :: thesis: ( not x in right_open_halfline a or a <= x )
thus ( not x in right_open_halfline a or a <= x ) by XXREAL_1:235; :: thesis: verum

end;

thus not right_open_halfline a is bounded_above by Lm2; :: thesis: right_open_halfline a is interval
let r, s be ExtReal; :: according to XXREAL_2:def 12 :: thesis: ( not r in right_open_halfline a or not s in right_open_halfline a or [.r,s.] c= right_open_halfline a )
assume A11: r in right_open_halfline a ; :: thesis: ( not s in right_open_halfline a or [.r,s.] c= right_open_halfline a )
then A12: a < r by XXREAL_1:235;
assume s in right_open_halfline a ; :: thesis: [.r,s.] c= right_open_halfline a
then reconsider rr = r, ss = s as Real by A11;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in [.r,s.] or x in right_open_halfline a )
assume A13: x in [.r,s.] ; :: thesis: x in right_open_halfline a
then x in [.rr,ss.] ;
then reconsider x = x as Real ;
r <= x by A13, XXREAL_1:1;
then a < x by A12, XXREAL_0:2;
hence x in right_open_halfline a by XXREAL_1:235; :: thesis: verum