let p be Point of (TOP-REAL 2); :: thesis: south_halfline p = { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) }
set A = { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) } ;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) } c= south_halfline p

let x be object ; :: thesis: ( x in south_halfline p implies x in { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) } )
assume A1: x in south_halfline p ; :: thesis: x in { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) }
then reconsider q = x as Point of (TOP-REAL 2) ;
A2: q `2 <= p `2 by A1, Def12;
q `1 = p `1 by A1, Def12;
hence x in { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) } by A2; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) } or x in south_halfline p )
assume x in { q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 ) } ; :: thesis: x in south_halfline p
then ex q being Point of (TOP-REAL 2) st
( x = q & q `1 = p `1 & q `2 <= p `2 ) ;
hence x in south_halfline p by Def12; :: thesis: verum