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 st

( r > 0 & Ball (p,r) c= (south_halfline a) ` )

then XX is open by Lm3, PRE_TOPC:30;

then XX ` is closed ;

hence south_halfline a is closed ; :: thesis: verum

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 st

( r > 0 & Ball (p,r) c= (south_halfline a) ` )

proof

then
OO is open
by TOPMETR:15;
let p be Point of (Euclid 2); :: thesis: ( p in (south_halfline a) ` implies ex r being Real st

( r > 0 & Ball (p,r) c= (south_halfline a) ` ) )

reconsider x = p as Point of (TOP-REAL 2) by EUCLID:67;

assume p in (south_halfline a) ` ; :: thesis: ex r being Real st

( r > 0 & Ball (p,r) c= (south_halfline a) ` )

then A17: not p in south_halfline a by XBOOLE_0:def 5;

end;( r > 0 & Ball (p,r) c= (south_halfline a) ` ) )

reconsider x = p as Point of (TOP-REAL 2) by EUCLID:67;

assume p in (south_halfline a) ` ; :: thesis: ex r being Real 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 12;

end;

suppose A18:
x `1 <> a `1
; :: thesis: ex r being Real st

( r > 0 & Ball (p,r) c= (south_halfline a) ` )

( r > 0 & Ball (p,r) c= (south_halfline a) ` )

take r = |.((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 COMPLEX1:47; :: thesis: Ball (p,r) c= (south_halfline a) `

let b be object ; :: 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:67;

dist (p,b) < r by A19, METRIC_1:11;

then A20: dist (x,c) < r by TOPREAL6:def 1;

hence b in (south_halfline a) ` by XBOOLE_0:def 5; :: thesis: verum

end;(x `1) - (a `1) <> 0 by A18;

hence r > 0 by COMPLEX1:47; :: thesis: Ball (p,r) c= (south_halfline a) `

let b be object ; :: 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:67;

dist (p,b) < r by A19, METRIC_1:11;

then A20: dist (x,c) < r by TOPREAL6:def 1;

now :: thesis: not c `1 = a `1

then
not c in south_halfline a
by TOPREAL1:def 12;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 A20, TOPREAL6:92;

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 A22, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < |.((x `1) - (c `1)).| ^2 by A21, SQUARE_1:16;

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 A22, SQUARE_1:def 2;

hence contradiction by A23, XREAL_1:7; :: thesis: verum

end;then A21: sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) < |.((x `1) - (c `1)).| by A20, TOPREAL6:92;

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 A22, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < |.((x `1) - (c `1)).| ^2 by A21, SQUARE_1:16;

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 A22, SQUARE_1:def 2;

hence contradiction by A23, XREAL_1:7; :: thesis: verum

hence b in (south_halfline a) ` by XBOOLE_0:def 5; :: thesis: verum

suppose A24:
x `2 > a `2
; :: thesis: ex r being Real st

( r > 0 & Ball (p,r) c= (south_halfline a) ` )

( 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_1:50; :: thesis: Ball (p,r) c= (south_halfline a) `

let b be object ; :: 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:67;

dist (p,b) < r by A25, METRIC_1:11;

then A26: dist (x,c) < r by TOPREAL6:def 1;

hence b in (south_halfline a) ` by XBOOLE_0:def 5; :: thesis: verum

end;thus r > 0 by A24, XREAL_1:50; :: thesis: Ball (p,r) c= (south_halfline a) `

let b be object ; :: 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:67;

dist (p,b) < r by A25, METRIC_1:11;

then A26: dist (x,c) < r by TOPREAL6:def 1;

now :: thesis: not c `2 <= a `2

then
not c in south_halfline a
by TOPREAL1:def 12;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 A24, XREAL_1:50;

then A28: ((x `2) - (a `2)) ^2 <= ((x `2) - (c `2)) ^2 by A27, SQUARE_1:15;

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 A26, TOPREAL6:92;

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 A29, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `2) - (a `2)) ^2 by A30, SQUARE_1:16;

then A32: (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < ((x `2) - (a `2)) ^2 by A29, A31, SQUARE_1:def 2;

0 + (((x `2) - (c `2)) ^2) <= (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) by A29, XREAL_1:7;

hence contradiction by A32, A28, XXREAL_0:2; :: thesis: verum

end;then A27: (x `2) - (c `2) >= (x `2) - (a `2) by XREAL_1:13;

0 <= (x `2) - (a `2) by A24, XREAL_1:50;

then A28: ((x `2) - (a `2)) ^2 <= ((x `2) - (c `2)) ^2 by A27, SQUARE_1:15;

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 A26, TOPREAL6:92;

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 A29, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `2) - (a `2)) ^2 by A30, SQUARE_1:16;

then A32: (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < ((x `2) - (a `2)) ^2 by A29, A31, SQUARE_1:def 2;

0 + (((x `2) - (c `2)) ^2) <= (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) by A29, XREAL_1:7;

hence contradiction by A32, A28, XXREAL_0:2; :: thesis: verum

hence b in (south_halfline a) ` by XBOOLE_0:def 5; :: thesis: verum

then XX is open by Lm3, PRE_TOPC:30;

then XX ` is closed ;

hence south_halfline a is closed ; :: thesis: verum