set X = north_halfline a;
reconsider XX = (north_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 (north_halfline a) ` holds
ex r being real number st
( r > 0 & Ball p,r c= (north_halfline a) ` )

proof

let p be Point of (Euclid 2); :: thesis: ( p in (north_halfline a) ` implies ex r being real number st
( r > 0 & Ball p,r c= (north_halfline a) ` ) )
reconsider x = p as Point of (TOP-REAL 2) by EUCLID:71;
assume p in (north_halfline a) ` ; :: thesis: ex r being real number st
( r > 0 & Ball p,r c= (north_halfline a) ` )
then A1: not p in north_halfline a by XBOOLE_0:def 5;

per cases ( x `1 <> a `1 or x `2 < a `2 ) by A1, TOPREAL1:def 12;

suppose A2: x `1 <> a `1 ; :: thesis: ex r being real number st
( r > 0 & Ball p,r c= (north_halfline a) ` )

take r = abs ((x `1 ) - (a `1 )); :: thesis: ( r > 0 & Ball p,r c= (north_halfline a) ` )
(x `1 ) - (a `1 ) <> 0 by A2;
hence r > 0 by COMPLEX1:133; :: thesis: Ball p,r c= (north_halfline a) `
let b be set ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball p,r or b in (north_halfline a) ` )
assume A3: b in Ball p,r ; :: thesis: b in (north_halfline a) `
then reconsider bb = b as Point of (Euclid 2) ;
reconsider c = bb as Point of (TOP-REAL 2) by EUCLID:71;
dist p,bb < r by A3, METRIC_1:12;
then A4: dist x,c < r by TOPREAL6:def 1;

now

assume c `1 = a `1 ; :: thesis: contradiction
then A5: sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) < abs ((x `1 ) - (c `1 )) by A4, TOPREAL6:101;
A6: 0 <= ((x `1 ) - (c `1 )) ^2 by XREAL_1:65;
A7: 0 <= ((x `2 ) - (c `2 )) ^2 by XREAL_1:65;
then 0 <= sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) by A6, SQUARE_1:def 4;
then (sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ))) ^2 < (abs ((x `1 ) - (c `1 ))) ^2 by A5, 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 A6, SQUARE_1:def 4;
hence contradiction by A7, XREAL_1:9; :: thesis: verum

end;

then not c in north_halfline a by TOPREAL1:def 12;
hence b in (north_halfline a) ` by XBOOLE_0:def 5; :: thesis: verum

end;

suppose A8: x `2 < a `2 ; :: thesis: ex r being real number st
( r > 0 & Ball p,r c= (north_halfline a) ` )

take r = (a `2 ) - (x `2 ); :: thesis: ( r > 0 & Ball p,r c= (north_halfline a) ` )
thus r > 0 by A8, XREAL_1:52; :: thesis: Ball p,r c= (north_halfline a) `
let b be set ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball p,r or b in (north_halfline a) ` )
assume A9: b in Ball p,r ; :: thesis: b in (north_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 A9, METRIC_1:12;
then A10: dist x,c < r by TOPREAL6:def 1;

now

assume c `2 >= a `2 ; :: thesis: contradiction
then A11: (a `2 ) - (x `2 ) <= (c `2 ) - (x `2 ) by XREAL_1:15;
0 <= (a `2 ) - (x `2 ) by A8, XREAL_1:52;
then A12: ((a `2 ) - (x `2 )) ^2 <= ((c `2 ) - (x `2 )) ^2 by A11, SQUARE_1:77;
A13: 0 <= ((x `1 ) - (c `1 )) ^2 by XREAL_1:65;
A14: sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) < (a `2 ) - (x `2 ) by A10, TOPREAL6:101;
A15: 0 <= ((x `2 ) - (c `2 )) ^2 by XREAL_1:65;
then 0 <= sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 )) by A13, SQUARE_1:def 4;
then (sqrt ((((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ))) ^2 < ((a `2 ) - (x `2 )) ^2 by A14, SQUARE_1:78;
then A16: (((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ) < ((a `2 ) - (x `2 )) ^2 by A13, A15, SQUARE_1:def 4;
0 + (((x `2 ) - (c `2 )) ^2 ) <= (((x `1 ) - (c `1 )) ^2 ) + (((x `2 ) - (c `2 )) ^2 ) by A13, XREAL_1:9;
hence contradiction by A16, A12, XXREAL_0:2; :: thesis: verum

end;

then not c in north_halfline a by TOPREAL1:def 12;
hence b in (north_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 north_halfline a is closed ; :: thesis: verum