set X = south_halfline a;
reconsider XX = () ` as Subset of () ;
reconsider OO = XX as Subset of () by Lm3;
for p being Point of () st p in () ` holds
ex r being Real st
( r > 0 & Ball (p,r) c= () ` )
proof
let p be Point of (); :: thesis: ( p in () ` implies ex r being Real st
( r > 0 & Ball (p,r) c= () ` ) )

reconsider x = p as Point of () by EUCLID:67;
assume p in () ` ; :: thesis: ex r being Real st
( r > 0 & Ball (p,r) c= () ` )

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

take r = |.((x `1) - (a `1)).|; :: thesis: ( r > 0 & Ball (p,r) c= () ` )
(x `1) - (a `1) <> 0 by A18;
hence r > 0 by COMPLEX1:47; :: thesis: Ball (p,r) c= () `
let b be object ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball (p,r) or b in () ` )
assume A19: b in Ball (p,r) ; :: thesis: b in () `
then reconsider b = b as Point of () ;
reconsider c = b as Point of () by EUCLID:67;
dist (p,b) < r by ;
then A20: dist (x,c) < r by TOPREAL6:def 1;
now :: thesis: not c `1 = a `1
assume c `1 = a `1 ; :: thesis: contradiction
then A21: sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) < |.((x `1) - (c `1)).| by ;
A22: 0 <= ((x `1) - (c `1)) ^2 by XREAL_1:63;
A23: 0 <= ((x `2) - (c `2)) ^2 by XREAL_1:63;
then 0 <= sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) by ;
then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < |.((x `1) - (c `1)).| ^2 by ;
then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `1) - (c `1)) ^2 by COMPLEX1:75;
then (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < (((x `1) - (c `1)) ^2) + 0 by ;
hence contradiction by A23, XREAL_1:7; :: thesis: verum
end;
then not c in south_halfline a by TOPREAL1:def 12;
hence b in () ` by XBOOLE_0:def 5; :: thesis: verum
end;
suppose A24: x `2 > a `2 ; :: thesis: ex r being Real st
( r > 0 & Ball (p,r) c= () ` )

take r = (x `2) - (a `2); :: thesis: ( r > 0 & Ball (p,r) c= () ` )
thus r > 0 by ; :: thesis: Ball (p,r) c= () `
let b be object ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball (p,r) or b in () ` )
assume A25: b in Ball (p,r) ; :: thesis: b in () `
then reconsider b = b as Point of () ;
reconsider c = b as Point of () by EUCLID:67;
dist (p,b) < r by ;
then A26: dist (x,c) < r by TOPREAL6:def 1;
now :: thesis: not c `2 <= a `2
assume c `2 <= a `2 ; :: thesis: contradiction
then A27: (x `2) - (c `2) >= (x `2) - (a `2) by XREAL_1:13;
0 <= (x `2) - (a `2) by ;
then A28: ((x `2) - (a `2)) ^2 <= ((x `2) - (c `2)) ^2 by ;
A29: 0 <= ((x `1) - (c `1)) ^2 by XREAL_1:63;
A30: sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) < (x `2) - (a `2) by ;
A31: 0 <= ((x `2) - (c `2)) ^2 by XREAL_1:63;
then 0 <= sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) by ;
then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `2) - (a `2)) ^2 by ;
then A32: (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < ((x `2) - (a `2)) ^2 by ;
0 + (((x `2) - (c `2)) ^2) <= (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) by ;
hence contradiction by A32, A28, XXREAL_0:2; :: thesis: verum
end;
then not c in south_halfline a by TOPREAL1:def 12;
hence b in () ` by XBOOLE_0:def 5; :: thesis: verum
end;
end;
end;
then OO is open by TOPMETR:15;
then XX is open by ;
then XX ` is closed ;
hence south_halfline a is closed ; :: thesis: verum