let G be Go-board; :: thesis: ( 1 < width G & 1 < len G implies LSeg ((G * 1,1) - |[1,1]|),(((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) c= ((Int (cell G,0 ,0 )) \/ (Int (cell G,1,0 ))) \/ {((G * 1,1) - |[0 ,1]|)} )
assume A1: ( 1 < width G & 1 < len G ) ; :: thesis: LSeg ((G * 1,1) - |[1,1]|),(((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) c= ((Int (cell G,0 ,0 )) \/ (Int (cell G,1,0 ))) \/ {((G * 1,1) - |[0 ,1]|)}
set q2 = G * 1,1;
set q3 = G * 2,1;
set r = 1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1);
A2: (0 + 1) + 1 = 0 + (1 + 1) ;
A3: 0 + (1 + 1) <= len G by A1, NAT_1:13;
then A4: (G * 1,1) `2 = (G * 2,1) `2 by A1, GOBOARD5:2;
(G * 1,1) `1 < (G * 2,1) `1 by A1, A3, GOBOARD5:4;
then ((G * 2,1) `1 ) - ((G * 1,1) `1 ) > 0 by XREAL_1:52;
then (1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 )) > 0 by XREAL_1:131;
then A6: 1 < ((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1 by XREAL_1:31;
then A7: 0 < 1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1) ;
A8: 1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1) < 1 by A6, XREAL_1:214;
A9: (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((1 / 2) * ((G * 2,1) `1 ))) - ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((1 / 2) * ((G * 1,1) `1 )))) + (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) = (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)
.= 1 by A6, XCMPLX_1:107 ;
A10: (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) - |[1,1]|)) `1 ) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `1 ) by Lm1
.= ((((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (G * 1,1)) - ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * |[1,1]|)) `1 ) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `1 ) by EUCLID:53
.= ((((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (G * 1,1)) `1 ) - (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * |[1,1]|) `1 )) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `1 ) by Lm2
.= ((((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (G * 1,1)) `1 ) - ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (|[1,1]| `1 ))) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `1 ) by Lm3
.= ((((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (G * 1,1)) `1 ) - ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `1 ) by EUCLID:56
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `1 )) - ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `1 ) by Lm3
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `1 )) - (1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) `1 )) by Lm3
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `1 )) - (1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) `1 ) - (|[0 ,1]| `1 ))) by Lm2
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `1 )) - (1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) `1 ) - 0 )) by EUCLID:56
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `1 )) - (1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((1 / 2) * (((G * 1,1) + (G * 2,1)) `1 ))) by Lm3
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `1 )) - (1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)))) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((1 / 2) * (((G * 1,1) `1 ) + ((G * 2,1) `1 )))) by Lm1
.= ((G * 1,1) `1 ) - 0 by A9
.= ((G * 1,1) `1 ) - (|[0 ,1]| `1 ) by EUCLID:56
.= ((G * 1,1) - |[0 ,1]|) `1 by Lm2 ;
(((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) - |[1,1]|)) `2 ) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `2 ) by Lm1
.= ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (((G * 1,1) - |[1,1]|) `2 )) + (((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) `2 ) by Lm3
.= ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (((G * 1,1) - |[1,1]|) `2 )) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) `2 )) by Lm3
.= ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (((G * 1,1) `2 ) - (|[1,1]| `2 ))) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) `2 )) by Lm2
.= ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (((G * 1,1) `2 ) - (|[1,1]| `2 ))) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) `2 ) - (|[0 ,1]| `2 ))) by Lm2
.= ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * (((G * 1,1) `2 ) - 1)) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) `2 ) - (|[0 ,1]| `2 ))) by EUCLID:56
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `2 )) - ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((((1 / 2) * ((G * 1,1) + (G * 2,1))) `2 ) - 1)) by EUCLID:56
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `2 )) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) `2 ))) - ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) + (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)))
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `2 )) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((1 / 2) * (((G * 1,1) + (G * 2,1)) `2 )))) - 1 by Lm3
.= (((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) `2 )) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * ((1 / 2) * (((G * 1,1) `2 ) + ((G * 1,1) `2 ))))) - 1 by A4, Lm1
.= ((G * 1,1) `2 ) - (|[0 ,1]| `2 ) by EUCLID:56
.= ((G * 1,1) - |[0 ,1]|) `2 by Lm2 ;
then ((1 - (1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1))) * ((G * 1,1) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * 2,1) `1 ) - ((G * 1,1) `1 ))) + 1)) * (((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) = |[(((G * 1,1) - |[0 ,1]|) `1 ),(((G * 1,1) - |[0 ,1]|) `2 )]| by A10, EUCLID:57
.= (G * 1,1) - |[0 ,1]| by EUCLID:57 ;
then (G * 1,1) - |[0 ,1]| in LSeg ((G * 1,1) - |[1,1]|),(((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) by A7, A8;
then A11: LSeg ((G * 1,1) - |[1,1]|),(((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) = (LSeg ((G * 1,1) - |[1,1]|),((G * 1,1) - |[0 ,1]|)) \/ (LSeg ((G * 1,1) - |[0 ,1]|),(((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|)) by A7, A8, TOPREAL1:11;
set I1 = Int (cell G,0 ,0 );
set I2 = Int (cell G,1,0 );
A12: LSeg ((G * 1,1) - |[1,1]|),((G * 1,1) - |[0 ,1]|) c= (Int (cell G,0 ,0 )) \/ {((G * 1,1) - |[0 ,1]|)} by Th63;
A13: LSeg ((G * 1,1) - |[0 ,1]|),(((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) c= (Int (cell G,1,0 )) \/ {((G * 1,1) - |[0 ,1]|)} by A1, A2, Th55;
((Int (cell G,0 ,0 )) \/ (Int (cell G,1,0 ))) \/ {((G * 1,1) - |[0 ,1]|)} = (Int (cell G,0 ,0 )) \/ ((Int (cell G,1,0 )) \/ ({((G * 1,1) - |[0 ,1]|)} \/ {((G * 1,1) - |[0 ,1]|)})) by XBOOLE_1:4
.= (Int (cell G,0 ,0 )) \/ (((Int (cell G,1,0 )) \/ {((G * 1,1) - |[0 ,1]|)}) \/ {((G * 1,1) - |[0 ,1]|)}) by XBOOLE_1:4
.= ((Int (cell G,0 ,0 )) \/ {((G * 1,1) - |[0 ,1]|)}) \/ ((Int (cell G,1,0 )) \/ {((G * 1,1) - |[0 ,1]|)}) by XBOOLE_1:4 ;
hence LSeg ((G * 1,1) - |[1,1]|),(((1 / 2) * ((G * 1,1) + (G * 2,1))) - |[0 ,1]|) c= ((Int (cell G,0 ,0 )) \/ (Int (cell G,1,0 ))) \/ {((G * 1,1) - |[0 ,1]|)} by A11, A12, A13, XBOOLE_1:13; :: thesis: verum