set A = left_open_halfline a;
thus not left_open_halfline a is bounded_below by Lm1; :: thesis: ( left_open_halfline a is bounded_above & left_open_halfline a is interval )
thus left_open_halfline a is bounded_above :: thesis: left_open_halfline a is interval

proof

take a ; :: according to XXREAL_2:def 10 :: thesis: a is UpperBound of left_open_halfline a
let x be ExtReal; :: according to XXREAL_2:def 1 :: thesis: ( not x in left_open_halfline a or x <= a )
thus ( not x in left_open_halfline a or x <= a ) by XXREAL_1:233; :: thesis: verum

end;

let r, s be ExtReal; :: according to XXREAL_2:def 12 :: thesis: ( not r in left_open_halfline a or not s in left_open_halfline a or [.r,s.] c= left_open_halfline a )
assume A4: r in left_open_halfline a ; :: thesis: ( not s in left_open_halfline a or [.r,s.] c= left_open_halfline a )
assume A5: s in left_open_halfline a ; :: thesis: [.r,s.] c= left_open_halfline a
then A6: s < a by XXREAL_1:233;
reconsider rr = r, ss = s as Real by A4, A5;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in [.r,s.] or x in left_open_halfline a )
assume A7: x in [.r,s.] ; :: thesis: x in left_open_halfline a
then x in [.rr,ss.] ;
then reconsider x = x as Real ;
x <= s by A7, XXREAL_1:1;
then x < a by A6, XXREAL_0:2;
hence x in left_open_halfline a by XXREAL_1:233; :: thesis: verum