let p be Point of (TOP-REAL 2); :: thesis: west_halfline p = { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 }
set A = { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 } ;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 } c= west_halfline p

let x be set ; :: thesis: ( x in west_halfline p implies x in { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 } )
assume A1: x in west_halfline p ; :: thesis: x in { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 }
then reconsider q = x as Point of (TOP-REAL 2) ;
A2: ( q `1 <= p `1 & q `2 = p `2 ) by A1, Def15;
q = |[(q `1 ),(q `2 )]| by EUCLID:57;
hence x in { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 } by A2; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 } or x in west_halfline p )
assume x in { |[r,(p `2 )]| where r is Element of REAL : r <= p `1 } ; :: thesis: x in west_halfline p
then consider r being Element of REAL such that
A3: x = |[r,(p `2 )]| and
A4: r <= p `1 ;
reconsider q = x as Point of (TOP-REAL 2) by A3;
( q `1 = r & q `2 = p `2 ) by A3, EUCLID:56;
hence x in west_halfline p by A4, Def15; :: thesis: verum