let G be Go-board; Int (h_strip (G,0)) = { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 }
0 <> len G
by MATRIX_0:def 10;
then
1 <= len G
by NAT_1:14;
then A1:
h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (1,1)) `2 }
by GOBOARD5:7;
thus
Int (h_strip (G,0)) c= { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 }
XBOOLE_0:def 10 { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } c= Int (h_strip (G,0))proof
let x be
object ;
TARSKI:def 3 ( not x in Int (h_strip (G,0)) or x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } )
assume A2:
x in Int (h_strip (G,0))
;
x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 }
then reconsider u =
x as
Point of
(Euclid 2) by Lm6;
consider s1 being
Real such that A3:
s1 > 0
and A4:
Ball (
u,
s1)
c= h_strip (
G,
0)
by A2, Th5;
reconsider p =
u as
Point of
(TOP-REAL 2) by Lm6;
A5:
p = |[(p `1),(p `2)]|
by EUCLID:53;
set q =
|[((p `1) + 0),((p `2) + (s1 / 2))]|;
s1 / 2
< s1
by A3, XREAL_1:216;
then
|[((p `1) + 0),((p `2) + (s1 / 2))]| in Ball (
u,
s1)
by A3, A5, Th8;
then
|[((p `1) + 0),((p `2) + (s1 / 2))]| in h_strip (
G,
0)
by A4;
then
ex
r2,
s2 being
Real st
(
|[((p `1) + 0),((p `2) + (s1 / 2))]| = |[r2,s2]| &
s2 <= (G * (1,1)) `2 )
by A1;
then A6:
(p `2) + (s1 / 2) <= (G * (1,1)) `2
by SPPOL_2:1;
p `2 < (p `2) + (s1 / 2)
by A3, XREAL_1:29, XREAL_1:215;
then
p `2 < (G * (1,1)) `2
by A6, XXREAL_0:2;
hence
x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 }
by A5;
verum
end;
let x be object ; TARSKI:def 3 ( not x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } or x in Int (h_strip (G,0)) )
assume
x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 }
; x in Int (h_strip (G,0))
then consider r, s being Real such that
A7:
x = |[r,s]|
and
A8:
s < (G * (1,1)) `2
;
reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:8;
A9:
Ball (u,(((G * (1,1)) `2) - s)) c= h_strip (G,0)
proof
let y be
object ;
TARSKI:def 3 ( not y in Ball (u,(((G * (1,1)) `2) - s)) or y in h_strip (G,0) )
A10:
Ball (
u,
(((G * (1,1)) `2) - s))
= { v where v is Point of (Euclid 2) : dist (u,v) < ((G * (1,1)) `2) - s }
by METRIC_1:17;
assume
y in Ball (
u,
(((G * (1,1)) `2) - s))
;
y in h_strip (G,0)
then consider v being
Point of
(Euclid 2) such that A11:
v = y
and A12:
dist (
u,
v)
< ((G * (1,1)) `2) - s
by A10;
reconsider q =
v as
Point of
(TOP-REAL 2) by TOPREAL3:8;
(
(s - (q `2)) ^2 >= 0 &
((s - (q `2)) ^2) + 0 <= ((r - (q `1)) ^2) + ((s - (q `2)) ^2) )
by XREAL_1:6, XREAL_1:63;
then A13:
sqrt ((s - (q `2)) ^2) <= sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2))
by SQUARE_1:26;
A14:
q = |[(q `1),(q `2)]|
by EUCLID:53;
then
sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) < ((G * (1,1)) `2) - s
by A12, Th6;
then
sqrt ((s - (q `2)) ^2) <= ((G * (1,1)) `2) - s
by A13, XXREAL_0:2;
then A15:
|.(s - (q `2)).| <= ((G * (1,1)) `2) - s
by COMPLEX1:72;
end;
reconsider B = Ball (u,(((G * (1,1)) `2) - s)) as Subset of (TOP-REAL 2) by TOPREAL3:8;
A17:
B is open
by Th3;
u in Ball (u,(((G * (1,1)) `2) - s))
by A8, Th1, XREAL_1:50;
hence
x in Int (h_strip (G,0))
by A7, A9, A17, TOPS_1:22; verum