let a be Point of (TOP-REAL 2); :: thesis: proj2 .: (south_halfline a) is bounded_above
take a `2 ; :: according to SEQ_4:def 1 :: thesis: for b<sub>1</sub> being set holds
( not b<sub>1</sub> in proj2 .: (south_halfline a) or b<sub>1</sub> <= a `2 )
let r be real number ; :: thesis: ( not r in proj2 .: (south_halfline a) or r <= a `2 )
assume r in proj2 .: (south_halfline a) ; :: thesis: r <= a `2
then consider x being set such that
A1: x in the carrier of (TOP-REAL 2) and
A2: x in south_halfline a and
A3: r = proj2 . x by FUNCT_2:115;
reconsider x = x as Point of (TOP-REAL 2) by A1;
r = x `2 by A3, PSCOMP_1:def 29;
hence r <= a `2 by A2, TOPREAL1:def 14; :: thesis: verum