let a be Point of (TOP-REAL 2); :: thesis: lower_bound (proj2 .: (north_halfline a)) = a `2
set X = proj2 .: (north_halfline a);

A1: now

let r be real number ; :: thesis: ( r in proj2 .: (north_halfline a) implies a `2 <= r )
assume r in proj2 .: (north_halfline a) ; :: thesis: a `2 <= r
then consider x being set such that
A2: x in the carrier of (TOP-REAL 2) and
A3: x in north_halfline a and
A4: r = proj2 . x by FUNCT_2:64;
reconsider x = x as Point of (TOP-REAL 2) by A2;
r = x `2 by A4, PSCOMP_1:def 6;
hence a `2 <= r by A3, TOPREAL1:def 10; :: thesis: verum

end;

A5: now

reconsider r = a `2 as real number ;
let s be real number ; :: thesis: ( 0 < s implies ex r being real number st
( r in proj2 .: (north_halfline a) & r < (a `2) + s ) )
assume 0 < s ; :: thesis: ex r being real number st
( r in proj2 .: (north_halfline a) & r < (a `2) + s )
then A6: r + 0 < (a `2) + s by XREAL_1:8;
take r = r; :: thesis: ( r in proj2 .: (north_halfline a) & r < (a `2) + s )
( a in north_halfline a & r = proj2 . a ) by PSCOMP_1:def 6, TOPREAL1:38;
hence r in proj2 .: (north_halfline a) by FUNCT_2:35; :: thesis: r < (a `2) + s
thus r < (a `2) + s by A6; :: thesis: verum

end;

proj2 .: (north_halfline a) is bounded_below by Th4;
hence lower_bound (proj2 .: (north_halfline a)) = a `2 by A1, A5, SEQ_4:def 2; :: thesis: verum