let G be Go-board; LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|)) c= (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)}
let x be object ; TARSKI:def 3 ( not x in LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|)) or x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} )
set r1 = (G * (1,1)) `1 ;
set s1 = (G * (1,(width G))) `2 ;
assume A1:
x in LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|))
; x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A2:
p = ((1 - r) * ((G * (1,(width G))) + |[(- 1),1]|)) + (r * ((G * (1,(width G))) + |[0,1]|))
and
0 <= r
and
A3:
r <= 1
by A1;
now ( ( r = 1 & p in {((G * (1,(width G))) + |[0,1]|)} ) or ( r < 1 & p in Int (cell (G,0,(width G))) ) )per cases
( r = 1 or r < 1 )
by A3, XXREAL_0:1;
case
r < 1
;
p in Int (cell (G,0,(width G)))then
1
- r > 0
by XREAL_1:50;
then
(G * (1,1)) `1 < ((G * (1,1)) `1) + (1 - r)
by XREAL_1:29;
then A4:
(
(G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + 1 &
((G * (1,1)) `1) - (1 - r) < (G * (1,1)) `1 )
by XREAL_1:19, XREAL_1:29;
0 <> width G
by MATRIX_0:def 10;
then A5:
1
<= width G
by NAT_1:14;
0 <> len G
by MATRIX_0:def 10;
then A6:
1
<= len G
by NAT_1:14;
A7:
G * (1,
(width G)) =
|[((G * (1,(width G))) `1),((G * (1,(width G))) `2)]|
by EUCLID:53
.=
|[((G * (1,1)) `1),((G * (1,(width G))) `2)]|
by A5, A6, GOBOARD5:2
;
A8:
Int (cell (G,0,(width G))) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s9 ) }
by Th19;
p =
(((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + (r * ((G * (1,(width G))) + |[0,1]|))
by A2, RLVECT_1:def 5
.=
(((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + ((r * (G * (1,(width G)))) + (r * |[0,1]|))
by RLVECT_1:def 5
.=
((r * (G * (1,(width G)))) + (((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|))) + (r * |[0,1]|)
by RLVECT_1:def 3
.=
(((r * (G * (1,(width G)))) + ((1 - r) * (G * (1,(width G))))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0,1]|)
by RLVECT_1:def 3
.=
(((r + (1 - r)) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0,1]|)
by RLVECT_1:def 6
.=
((G * (1,(width G))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0,1]|)
by RLVECT_1:def 8
.=
((G * (1,(width G))) + |[((1 - r) * (- 1)),((1 - r) * 1)]|) + (r * |[0,1]|)
by EUCLID:58
.=
((G * (1,(width G))) + |[(r - 1),(1 - r)]|) + |[(r * 0),(r * 1)]|
by EUCLID:58
.=
|[(((G * (1,1)) `1) + (r - 1)),(((G * (1,(width G))) `2) + (1 - r))]| + |[0,r]|
by A7, EUCLID:56
.=
|[((((G * (1,1)) `1) + (r - 1)) + 0),((((G * (1,(width G))) `2) + (1 - r)) + r)]|
by EUCLID:56
.=
|[(((G * (1,1)) `1) - (1 - r)),(((G * (1,(width G))) `2) + 1)]|
;
hence
p in Int (cell (G,0,(width G)))
by A4, A8;
verum end; end; end;
hence
x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)}
by XBOOLE_0:def 3; verum