set X = south_halfline a;
reconsider XX = (south_halfline a) ` as Subset of (TOP-REAL 2) ;
reconsider OO = XX as Subset of (TopSpaceMetr (Euclid 2)) by Lm3;
for p being Point of (Euclid 2) st p in (south_halfline a) ` holds
ex r being real number st
( r > 0 & Ball p,r c= (south_halfline a) ` )

proof

let p be Point of (Euclid 2); :: thesis: ( p in (south_halfline a) ` implies ex r being real number st
( r > 0 & Ball p,r c= (south_halfline a) ` ) )
reconsider x = p as Point of (TOP-REAL 2) by EUCLID:71;
assume p in (south_halfline a) ` ; :: thesis: ex r being real number st
( r > 0 & Ball p,r c= (south_halfline a) ` )
then A17: not p in south_halfline a by XBOOLE_0:def 5;

per cases ( x `1 <> a `1 or x `2 > a `2 ) by A17, TOPREAL1:def 14;

suppose A18: x `1 <> a `1 ; :: thesis: ex r being real number st
( r > 0 & Ball p,r c= (south_halfline a) ` )

take r = abs ((x `1 ) - (a `1 )); :: thesis: ( r > 0 & Ball p,r c= (south_halfline a) ` )
(x `1 ) - (a `1 ) <> 0 by A18;
hence r > 0 by COMPLEX1:133; :: thesis: Ball p,r c= (south_halfline a) `
let b be set ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball p,r or b in (south_halfline a) ` )
assume A19: b in Ball p,r ; :: thesis: b in (south_halfline a) `
then reconsider b = b as Point of (Euclid 2) ;
reconsider c = b as Point of (TOP-REAL 2) by EUCLID:71;
dist p,b < r by A19, METRIC_1:12;
then A20: dist x,c < r by TOPREAL6:def 1;

now

assume c `1 = a `1 ; :: thesis: contradiction
then A21: sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) < abs ((x `1 ) - (c `1 )) by A20, TOPREAL6:101;
A22: 0 <= ((x `1 ) - (c `1 )) ^2 by XREAL_1:65;
A23: 0 <= ((x `2 ) - (c `2 )) ^2 by XREAL_1:65;
then 0 <= sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) by A22, SQUARE_1:def 4;
then (sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ))) ^2 < (abs ((x `1 ) - (c `1 ))) ^2 by A21, SQUARE_1:78;
then (sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ))) ^2 < ((x `1 ) - (c `1 )) ^2 by COMPLEX1:161;
then (((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ) < (((x `1 ) - (c `1 )) ^2 ) + 0 by A22, SQUARE_1:def 4;
hence contradiction by A23, XREAL_1:9; :: thesis: verum

end;

then not c in south_halfline a by TOPREAL1:def 14;
hence b in (south_halfline a) ` by XBOOLE_0:def 5; :: thesis: verum

end;

suppose A24: x `2 > a `2 ; :: thesis: ex r being real number st
( r > 0 & Ball p,r c= (south_halfline a) ` )

take r = (x `2 ) - (a `2 ); :: thesis: ( r > 0 & Ball p,r c= (south_halfline a) ` )
thus r > 0 by A24, XREAL_1:52; :: thesis: Ball p,r c= (south_halfline a) `
let b be set ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball p,r or b in (south_halfline a) ` )
assume A25: b in Ball p,r ; :: thesis: b in (south_halfline a) `
then reconsider b = b as Point of (Euclid 2) ;
reconsider c = b as Point of (TOP-REAL 2) by EUCLID:71;
dist p,b < r by A25, METRIC_1:12;
then A26: dist x,c < r by TOPREAL6:def 1;

now

assume c `2 <= a `2 ; :: thesis: contradiction
then A27: (x `2 ) - (c `2 ) >= (x `2 ) - (a `2 ) by XREAL_1:15;
0 <= (x `2 ) - (a `2 ) by A24, XREAL_1:52;
then A28: ((x `2 ) - (a `2 )) ^2 <= ((x `2 ) - (c `2 )) ^2 by A27, SQUARE_1:77;
A29: 0 <= ((x `1 ) - (c `1 )) ^2 by XREAL_1:65;
A30: sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) < (x `2 ) - (a `2 ) by A26, TOPREAL6:101;
A31: 0 <= ((x `2 ) - (c `2 )) ^2 by XREAL_1:65;
then 0 <= sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) by A29, SQUARE_1:def 4;
then (sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ))) ^2 < ((x `2 ) - (a `2 )) ^2 by A30, SQUARE_1:78;
then A32: (((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ) < ((x `2 ) - (a `2 )) ^2 by A29, A31, SQUARE_1:def 4;
0 + (((x `2 ) - (c `2 )) ^2 ) <= (((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ) by A29, XREAL_1:9;
hence contradiction by A32, A28, XXREAL_0:2; :: thesis: verum

end;

then not c in south_halfline a by TOPREAL1:def 14;
hence b in (south_halfline a) ` by XBOOLE_0:def 5; :: thesis: verum

end;

end;

end;

then OO is open by TOPMETR:22;
then XX is open by Lm3, PRE_TOPC:60;
then XX ` is closed ;
hence south_halfline a is closed ; :: thesis: verum