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

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

let x be object ; :: thesis: ( x in west_halfline p implies x in { |[r,(p `2)]| where r is Real : r <= p `1 } )
assume A1: x in west_halfline p ; :: thesis: x in { |[r,(p `2)]| where r is Real : r <= p `1 }
then reconsider q = x as Point of (TOP-REAL 2) ;
A2: q `2 = p `2 by A1, Def13;
A3: q = |[(q `1),(q `2)]| by EUCLID:53;
q `1 <= p `1 by A1, Def13;
hence x in { |[r,(p `2)]| where r is Real : r <= p `1 } by A2, A3; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[r,(p `2)]| where r is Real : r <= p `1 } or x in west_halfline p )
assume x in { |[r,(p `2)]| where r is Real : r <= p `1 } ; :: thesis: x in west_halfline p
then consider r being Real such that
A4: x = |[r,(p `2)]| and
A5: r <= p `1 ;
reconsider q = x as Point of (TOP-REAL 2) by A4;
A6: q `2 = p `2 by A4, EUCLID:52;
q `1 = r by A4, EUCLID:52;
hence x in west_halfline p by A5, A6, Def13; :: thesis: verum