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

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

let x be set ; :: thesis: ( x in south_halfline p implies x in { |[(p `1 ),r]| where r is Element of REAL : r <= p `2 } )
assume A1: x in south_halfline p ; :: thesis: x in { |[(p `1 ),r]| where r is Element of REAL : r <= p `2 }
then reconsider q = x as Point of (TOP-REAL 2) ;
A2: ( q `1 = p `1 & q `2 <= p `2 ) by A1, Def14;
q = |[(q `1 ),(q `2 )]| by EUCLID:57;
hence x in { |[(p `1 ),r]| where r is Element of REAL : r <= p `2 } by A2; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[(p `1 ),r]| where r is Element of REAL : r <= p `2 } or x in south_halfline p )
assume x in { |[(p `1 ),r]| where r is Element of REAL : r <= p `2 } ; :: thesis: x in south_halfline p
then consider r being Element of REAL such that
A3: x = |[(p `1 ),r]| and
A4: r <= p `2 ;
reconsider q = x as Point of (TOP-REAL 2) by A3;
( q `1 = p `1 & q `2 = r ) by A3, EUCLID:56;
hence x in south_halfline p by A4, Def14; :: thesis: verum