set X = west_halfline a;

reconsider XX = (west_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 (west_halfline a) ` holds

ex r being Real st

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

then XX is open by Lm3, PRE_TOPC:30;

then XX ` is closed ;

hence west_halfline a is closed ; :: thesis: verum

reconsider XX = (west_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 (west_halfline a) ` holds

ex r being Real st

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

proof

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

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

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

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

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

then A49: not p in west_halfline a by XBOOLE_0:def 5;

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

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

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

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

then A49: not p in west_halfline a by XBOOLE_0:def 5;

per cases
( x `2 <> a `2 or x `1 > a `1 )
by A49, TOPREAL1:def 13;

end;

suppose A50:
x `2 <> a `2
; :: thesis: ex r being Real st

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

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

take r = |.((x `2) - (a `2)).|; :: thesis: ( r > 0 & Ball (p,r) c= (west_halfline a) ` )

(x `2) - (a `2) <> 0 by A50;

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

let b be object ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball (p,r) or b in (west_halfline a) ` )

assume A51: b in Ball (p,r) ; :: thesis: b in (west_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 A51, METRIC_1:11;

then A52: dist (x,c) < r by TOPREAL6:def 1;

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

end;(x `2) - (a `2) <> 0 by A50;

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

let b be object ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball (p,r) or b in (west_halfline a) ` )

assume A51: b in Ball (p,r) ; :: thesis: b in (west_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 A51, METRIC_1:11;

then A52: dist (x,c) < r by TOPREAL6:def 1;

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

then
not c in west_halfline a
by TOPREAL1:def 13;assume
c `2 = a `2
; :: thesis: contradiction

then A53: sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) < |.((x `2) - (c `2)).| by A52, TOPREAL6:92;

A54: 0 <= ((x `2) - (c `2)) ^2 by XREAL_1:63;

A55: 0 <= ((x `1) - (c `1)) ^2 by XREAL_1:63;

then 0 <= sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) by A54, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < |.((x `2) - (c `2)).| ^2 by A53, SQUARE_1:16;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `2) - (c `2)) ^2 by COMPLEX1:75;

then (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < (((x `2) - (c `2)) ^2) + 0 by A54, SQUARE_1:def 2;

hence contradiction by A55, XREAL_1:7; :: thesis: verum

end;then A53: sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) < |.((x `2) - (c `2)).| by A52, TOPREAL6:92;

A54: 0 <= ((x `2) - (c `2)) ^2 by XREAL_1:63;

A55: 0 <= ((x `1) - (c `1)) ^2 by XREAL_1:63;

then 0 <= sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) by A54, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < |.((x `2) - (c `2)).| ^2 by A53, SQUARE_1:16;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `2) - (c `2)) ^2 by COMPLEX1:75;

then (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < (((x `2) - (c `2)) ^2) + 0 by A54, SQUARE_1:def 2;

hence contradiction by A55, XREAL_1:7; :: thesis: verum

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

suppose A56:
x `1 > a `1
; :: thesis: ex r being Real st

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

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

take r = (x `1) - (a `1); :: thesis: ( r > 0 & Ball (p,r) c= (west_halfline a) ` )

thus r > 0 by A56, XREAL_1:50; :: thesis: Ball (p,r) c= (west_halfline a) `

let b be object ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball (p,r) or b in (west_halfline a) ` )

assume A57: b in Ball (p,r) ; :: thesis: b in (west_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 A57, METRIC_1:11;

then A58: dist (x,c) < r by TOPREAL6:def 1;

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

end;thus r > 0 by A56, XREAL_1:50; :: thesis: Ball (p,r) c= (west_halfline a) `

let b be object ; :: according to TARSKI:def 3 :: thesis: ( not b in Ball (p,r) or b in (west_halfline a) ` )

assume A57: b in Ball (p,r) ; :: thesis: b in (west_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 A57, METRIC_1:11;

then A58: dist (x,c) < r by TOPREAL6:def 1;

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

then
not c in west_halfline a
by TOPREAL1:def 13;assume
c `1 <= a `1
; :: thesis: contradiction

then A59: (x `1) - (c `1) >= (x `1) - (a `1) by XREAL_1:13;

0 <= (x `1) - (a `1) by A56, XREAL_1:50;

then A60: ((x `1) - (a `1)) ^2 <= ((x `1) - (c `1)) ^2 by A59, SQUARE_1:15;

A61: 0 <= ((x `2) - (c `2)) ^2 by XREAL_1:63;

A62: sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) < (x `1) - (a `1) by A58, TOPREAL6:92;

A63: 0 <= ((x `1) - (c `1)) ^2 by XREAL_1:63;

then 0 <= sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) by A61, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `1) - (a `1)) ^2 by A62, SQUARE_1:16;

then A64: (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < ((x `1) - (a `1)) ^2 by A61, A63, SQUARE_1:def 2;

0 + (((x `1) - (c `1)) ^2) <= (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) by A61, XREAL_1:7;

hence contradiction by A64, A60, XXREAL_0:2; :: thesis: verum

end;then A59: (x `1) - (c `1) >= (x `1) - (a `1) by XREAL_1:13;

0 <= (x `1) - (a `1) by A56, XREAL_1:50;

then A60: ((x `1) - (a `1)) ^2 <= ((x `1) - (c `1)) ^2 by A59, SQUARE_1:15;

A61: 0 <= ((x `2) - (c `2)) ^2 by XREAL_1:63;

A62: sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) < (x `1) - (a `1) by A58, TOPREAL6:92;

A63: 0 <= ((x `1) - (c `1)) ^2 by XREAL_1:63;

then 0 <= sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2)) by A61, SQUARE_1:def 2;

then (sqrt ((((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2))) ^2 < ((x `1) - (a `1)) ^2 by A62, SQUARE_1:16;

then A64: (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) < ((x `1) - (a `1)) ^2 by A61, A63, SQUARE_1:def 2;

0 + (((x `1) - (c `1)) ^2) <= (((x `1) - (c `1)) ^2) + (((x `2) - (c `2)) ^2) by A61, XREAL_1:7;

hence contradiction by A64, A60, XXREAL_0:2; :: thesis: verum

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

then XX is open by Lm3, PRE_TOPC:30;

then XX ` is closed ;

hence west_halfline a is closed ; :: thesis: verum