set A = left_closed_halfline a;
not left_open_halfline a is bounded_below by Lm1;
hence not left_closed_halfline a is bounded_below by XXREAL_1:21, XXREAL_2:44; :: thesis: ( left_closed_halfline a is bounded_above & left_closed_halfline a is connected )
thus left_closed_halfline a is bounded_above :: thesis: left_closed_halfline a is connected

proof

take a ; :: according to SEQ_4:def 1 :: thesis: for b₁ being set holds
( not b₁ in left_closed_halfline a or b₁ <= a )
thus for b₁ being set holds
( not b₁ in left_closed_halfline a or b₁ <= a ) by XXREAL_1:234; :: thesis: verum

end;

let r be real number ; :: according to JCT_MISC:def 1 :: thesis: for b₁ being set holds
( not r in left_closed_halfline a or not b₁ in left_closed_halfline a or [.r,b₁.] c= left_closed_halfline a )
let s be real number ; :: thesis: ( not r in left_closed_halfline a or not s in left_closed_halfline a or [.r,s.] c= left_closed_halfline a )
assume r in left_closed_halfline a ; :: thesis: ( not s in left_closed_halfline a or [.r,s.] c= left_closed_halfline a )
assume s in left_closed_halfline a ; :: thesis: [.r,s.] c= left_closed_halfline a
then A1: s <= a by XXREAL_1:234;
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in [.r,s.] or x in left_closed_halfline a )
assume A2: x in [.r,s.] ; :: thesis: x in left_closed_halfline a
then reconsider x = x as Real ;
x <= s by A2, XXREAL_1:1;
then x <= a by A1, XXREAL_0:2;
hence x in left_closed_halfline a by XXREAL_1:234; :: thesis: verum