let G be Go-board; ( 1 < len G & 1 < width G implies LSeg ((G * 1,(width G)) + |[(- 1),1]|),(((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) c= ((Int (cell G,0 ,(width G))) \/ (Int (cell G,0 ,((width G) -' 1)))) \/ {((G * 1,(width G)) - |[1,0 ]|)} )
assume that
A1:
1 < len G
and
A2:
1 < width G
; LSeg ((G * 1,(width G)) + |[(- 1),1]|),(((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) c= ((Int (cell G,0 ,(width G))) \/ (Int (cell G,0 ,((width G) -' 1)))) \/ {((G * 1,(width G)) - |[1,0 ]|)}
set q2 = G * 1,(width G);
set q3 = G * 1,((width G) -' 1);
set r = 1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1);
A3:
((width G) -' 1) + 1 = width G
by A2, XREAL_1:237;
then A4:
(width G) -' 1 >= 1
by A2, NAT_1:13;
A5:
(width G) -' 1 < width G
by A3, NAT_1:13;
then
(G * 1,((width G) -' 1)) `2 < (G * 1,(width G)) `2
by A1, A4, GOBOARD5:5;
then A6:
((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ) > 0
by XREAL_1:52;
then
1 < ((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1
by XREAL_1:31, XREAL_1:131;
then A7:
1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1) < 1
by XREAL_1:214;
A8: (G * 1,(width G)) `1 =
(G * 1,1) `1
by A1, A2, GOBOARD5:3
.=
(G * 1,((width G) -' 1)) `1
by A1, A4, A5, GOBOARD5:3
;
A9: (((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|))) `1 =
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) + |[(- 1),1]|)) `1 ) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `1 )
by Lm1
.=
((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (((G * 1,(width G)) + |[(- 1),1]|) `1 )) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `1 )
by Lm3
.=
((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (((G * 1,(width G)) + |[(- 1),1]|) `1 )) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) `1 ))
by Lm3
.=
((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (((G * 1,(width G)) `1 ) + (|[(- 1),1]| `1 ))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) `1 ))
by Lm1
.=
((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (((G * 1,(width G)) `1 ) + (|[(- 1),1]| `1 ))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) `1 ) - (|[1,0 ]| `1 )))
by Lm2
.=
((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (((G * 1,(width G)) `1 ) + (- 1))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) `1 ) - (|[1,0 ]| `1 )))
by EUCLID:56
.=
((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (((G * 1,(width G)) `1 ) - 1)) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) `1 ) - 1))
by EUCLID:56
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `1 )) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) `1 ))) - 1
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `1 )) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (((G * 1,(width G)) + (G * 1,((width G) -' 1))) `1 )))) - 1
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `1 )) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (((G * 1,(width G)) `1 ) + ((G * 1,(width G)) `1 ))))) - 1
by A8, Lm1
.=
((G * 1,(width G)) `1 ) - (|[1,0 ]| `1 )
by EUCLID:56
.=
((G * 1,(width G)) - |[1,0 ]|) `1
by Lm2
;
A10: (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * 1,(width G)) `2 ))) - ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * 1,((width G) -' 1)) `2 )))) + (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) =
(1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)
.=
1
by A6, XCMPLX_1:107
;
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|))) `2 =
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) + |[(- 1),1]|)) `2 ) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `2 )
by Lm1
.=
((((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (G * 1,(width G))) + ((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * |[(- 1),1]|)) `2 ) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `2 )
by EUCLID:36
.=
((((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (G * 1,(width G))) `2 ) + (((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * |[(- 1),1]|) `2 )) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `2 )
by Lm1
.=
((((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (G * 1,(width G))) `2 ) + ((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (|[(- 1),1]| `2 ))) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `2 )
by Lm3
.=
((((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * (G * 1,(width G))) `2 ) + ((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `2 )
by EUCLID:56
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `2 )) + ((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) `2 )
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `2 )) + (1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) `2 ))
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `2 )) + (1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) `2 ) - (|[1,0 ]| `2 )))
by Lm2
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `2 )) + (1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) `2 ) - 0 ))
by EUCLID:56
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `2 )) + (1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (((G * 1,(width G)) + (G * 1,((width G) -' 1))) `2 )))
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) `2 )) + (1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (((G * 1,(width G)) `2 ) + ((G * 1,((width G) -' 1)) `2 ))))
by Lm1
.=
((G * 1,(width G)) `2 ) - 0
by A10
.=
((G * 1,(width G)) `2 ) - (|[1,0 ]| `2 )
by EUCLID:56
.=
((G * 1,(width G)) - |[1,0 ]|) `2
by Lm2
;
then ((1 - (1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1))) * ((G * 1,(width G)) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)) =
|[(((G * 1,(width G)) - |[1,0 ]|) `1 ),(((G * 1,(width G)) - |[1,0 ]|) `2 )]|
by A9, EUCLID:57
.=
(G * 1,(width G)) - |[1,0 ]|
by EUCLID:57
;
then
(G * 1,(width G)) - |[1,0 ]| in LSeg ((G * 1,(width G)) + |[(- 1),1]|),(((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|)
by A6, A7;
then A11:
LSeg ((G * 1,(width G)) + |[(- 1),1]|),(((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) = (LSeg ((G * 1,(width G)) + |[(- 1),1]|),((G * 1,(width G)) - |[1,0 ]|)) \/ (LSeg ((G * 1,(width G)) - |[1,0 ]|),(((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|))
by TOPREAL1:11;
set I1 = Int (cell G,0 ,(width G));
set I2 = Int (cell G,0 ,((width G) -' 1));
A12: ((Int (cell G,0 ,(width G))) \/ (Int (cell G,0 ,((width G) -' 1)))) \/ {((G * 1,(width G)) - |[1,0 ]|)} =
(Int (cell G,0 ,(width G))) \/ ((Int (cell G,0 ,((width G) -' 1))) \/ ({((G * 1,(width G)) - |[1,0 ]|)} \/ {((G * 1,(width G)) - |[1,0 ]|)}))
by XBOOLE_1:4
.=
(Int (cell G,0 ,(width G))) \/ (((Int (cell G,0 ,((width G) -' 1))) \/ {((G * 1,(width G)) - |[1,0 ]|)}) \/ {((G * 1,(width G)) - |[1,0 ]|)})
by XBOOLE_1:4
.=
((Int (cell G,0 ,(width G))) \/ {((G * 1,(width G)) - |[1,0 ]|)}) \/ ((Int (cell G,0 ,((width G) -' 1))) \/ {((G * 1,(width G)) - |[1,0 ]|)})
by XBOOLE_1:4
;
A13:
LSeg ((G * 1,(width G)) + |[(- 1),1]|),((G * 1,(width G)) - |[1,0 ]|) c= (Int (cell G,0 ,(width G))) \/ {((G * 1,(width G)) - |[1,0 ]|)}
by Th61;
LSeg ((G * 1,(width G)) - |[1,0 ]|),(((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) c= (Int (cell G,0 ,((width G) -' 1))) \/ {((G * 1,(width G)) - |[1,0 ]|)}
by A3, A4, A5, Th52;
hence
LSeg ((G * 1,(width G)) + |[(- 1),1]|),(((1 / 2) * ((G * 1,(width G)) + (G * 1,((width G) -' 1)))) - |[1,0 ]|) c= ((Int (cell G,0 ,(width G))) \/ (Int (cell G,0 ,((width G) -' 1)))) \/ {((G * 1,(width G)) - |[1,0 ]|)}
by A11, A13, A12, XBOOLE_1:13; verum