set A = right_closed_halfline a;
thus right_closed_halfline a is bounded_below ; :: thesis: ( not right_closed_halfline a is bounded_above & right_closed_halfline a is interval )
not right_open_halfline a is bounded_above by Lm2;
hence not right_closed_halfline a is bounded_above by XXREAL_1:22, XXREAL_2:43; :: thesis: right_closed_halfline a is interval
let r, s be ExtReal; :: according to XXREAL_2:def 12 :: thesis: ( not r in right_closed_halfline a or not s in right_closed_halfline a or [.r,s.] c= right_closed_halfline a )
assume A8: r in right_closed_halfline a ; :: thesis: ( not s in right_closed_halfline a or [.r,s.] c= right_closed_halfline a )
then A9: a <= r by XXREAL_1:236;
assume s in right_closed_halfline a ; :: thesis: [.r,s.] c= right_closed_halfline a
then reconsider rr = r, ss = s as Real by A8;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in [.r,s.] or x in right_closed_halfline a )
assume A10: x in [.r,s.] ; :: thesis: x in right_closed_halfline a
then x in [.rr,ss.] ;
then reconsider x = x as Real ;
r <= x by A10, XXREAL_1:1;
then a <= x by A9, XXREAL_0:2;
hence x in right_closed_halfline a by XXREAL_1:236; :: thesis: verum