let G be Go-board; :: thesis:
assume A1: ( 1 < len G & 1 < width G ) ; :: thesis:
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);
A2: ((width G) -' 1) + 1 = width G by A1, XREAL_1:237;
then A3: (width G) -' 1 >= 1 by A1, NAT_1:13;
A4: (width G) -' 1 < width G by A2, NAT_1:13;
A5: (G * 1,(width G)) `1 = (G * 1,1) `1 by A1, GOBOARD5:3
.= (G * 1,((width G) -' 1)) `1 by A1, A3, A4, GOBOARD5:3 ;
(G * 1,((width G) -' 1)) `2 < (G * 1,(width G)) `2 by A1, A3, A4, GOBOARD5:5;
then ((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ) > 0 by XREAL_1:52;
then (1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 )) > 0 by XREAL_1:131;
then A7: 1 < ((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1 by XREAL_1:31;
then A8: 0 < 1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1) ;
A9: 1 / (((1 / 2) * (((G * 1,(width G)) `2 ) - ((G * 1,((width G) -' 1)) `2 ))) + 1) < 1 by A7, XREAL_1:214;
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 A7, XCMPLX_1:107 ;
A11: (((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 ;
(((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 A5, Lm1
.= ((G * 1,(width G)) `1 ) - (|[1,0 ]| `1 ) by EUCLID:56
.= ((G * 1,(width G)) - |[1,0 ]|) `1 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 A11, 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 A8, A9;
then A12: 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 A8, A9, TOPREAL1:11;
set I1 = Int (cell G,0 ,(width G));
set I2 = Int (cell G,0 ,((width G) -' 1));
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;
A14: 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 A2, A3, A4, Th52;
((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 ;
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 A12, A13, A14, XBOOLE_1:13; :: thesis: