let i be Element of NAT ; for G being Go-board st 1 <= i & i < len G & 1 < width G holds
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))) c= ((Int (cell G,i,((width G) -' 1))) \/ (Int (cell G,i,(width G)))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}
let G be Go-board; ( 1 <= i & i < len G & 1 < width G implies LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))) c= ((Int (cell G,i,((width G) -' 1))) \/ (Int (cell G,i,(width G)))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))} )
assume that
A1:
1 <= i
and
A2:
i < len G
and
A3:
1 < width G
; LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))) c= ((Int (cell G,i,((width G) -' 1))) \/ (Int (cell G,i,(width G)))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}
set I1 = Int (cell G,i,((width G) -' 1));
set I2 = Int (cell G,i,(width G));
set p1 = G * i,(width G);
set q2 = G * (i + 1),(width G);
set q3 = G * (i + 1),((width G) -' 1);
set r = 1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1);
A4:
((width G) -' 1) + 1 = width G
by A3, XREAL_1:237;
then A5:
1 <= (width G) -' 1
by A3, NAT_1:13;
A6:
(width G) -' 1 < width G
by A4, NAT_1:13;
then
(G * i,(width G)) + (G * (i + 1),((width G) -' 1)) = (G * i,((width G) -' 1)) + (G * (i + 1),(width G))
by A1, A2, A4, A5, Th14;
then A7:
LSeg ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))) c= (Int (cell G,i,((width G) -' 1))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}
by A1, A2, A4, A5, A6, Th44;
A8:
( i + 1 >= 1 & i + 1 <= len G )
by A2, NAT_1:11, NAT_1:13;
then A9: (G * (i + 1),(width G)) `1 =
(G * (i + 1),1) `1
by A3, GOBOARD5:3
.=
(G * (i + 1),((width G) -' 1)) `1
by A5, A6, A8, GOBOARD5:3
;
A10: (G * (i + 1),(width G)) `2 =
(G * 1,(width G)) `2
by A3, A8, GOBOARD5:2
.=
(G * i,(width G)) `2
by A1, A2, A3, GOBOARD5:2
;
then
(G * (i + 1),((width G) -' 1)) `2 < (G * i,(width G)) `2
by A5, A6, A8, GOBOARD5:5;
then A11:
((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ) > 0
by XREAL_1:52;
then
1 < ((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1
by XREAL_1:31, XREAL_1:131;
then A12:
1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1) < 1
by XREAL_1:214;
A13: ((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|)) `1 =
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) `1 ) + (((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|) `1 )
by Lm1
.=
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) `1 ) + (|[((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * 0 ),((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * 1)]| `1 )
by EUCLID:62
.=
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) `1 ) + 0
by EUCLID:56
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) `1 ) + (((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1)))) `1 )
by Lm1
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * (((1 / 2) * (G * (i + 1),(width G))) `1 )) + (((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1)))) `1 )
by Lm3
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * (((1 / 2) * (G * (i + 1),(width G))) `1 )) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (G * (i + 1),((width G) -' 1))) `1 ))
by Lm3
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * ((G * (i + 1),(width G)) `1 ))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (G * (i + 1),((width G) -' 1))) `1 ))
by Lm3
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * ((G * (i + 1),(width G)) `1 ))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * (i + 1),(width G)) `1 )))
by A9, Lm3
.=
((1 / 2) * (G * (i + 1),(width G))) `1
by Lm3
;
A14: ((Int (cell G,i,((width G) -' 1))) \/ (Int (cell G,i,(width G)))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))} =
(Int (cell G,i,((width G) -' 1))) \/ ((Int (cell G,i,(width G))) \/ ({((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))} \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}))
by XBOOLE_1:4
.=
(Int (cell G,i,((width G) -' 1))) \/ (((Int (cell G,i,(width G))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))})
by XBOOLE_1:4
.=
((Int (cell G,i,((width G) -' 1))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}) \/ ((Int (cell G,i,(width G))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))})
by XBOOLE_1:4
;
A15: ((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * i,(width G))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1)))) =
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * i,(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))
by EUCLID:30
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G))))) + (((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * i,(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1)))))
by EUCLID:30
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (G * i,(width G))) + ((1 / 2) * (G * (i + 1),((width G) -' 1)))))
by EUCLID:36
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * (((1 / 2) * (G * i,(width G))) + ((1 / 2) * (G * (i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (G * i,(width G))) + ((1 / 2) * (G * (i + 1),((width G) -' 1)))))
by EUCLID:36
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * (((1 / 2) * (G * i,(width G))) + ((1 / 2) * (G * (i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))))
by EUCLID:36
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))))
by EUCLID:36
;
A16: (((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * (i + 1),(width G)) `2 ))) - ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * (i + 1),((width G) -' 1)) `2 )))) + (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) =
(1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (((G * (i + 1),(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)
.=
1
by A10, A11, XCMPLX_1:107
;
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|)) `2 =
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) `2 ) + (((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|) `2 )
by Lm1
.=
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) `2 ) + (|[((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * 0 ),((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * 1)]| `2 )
by EUCLID:62
.=
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) `2 ) + (1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)))
by EUCLID:56
.=
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) `2 ) + (((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1)))) `2 )) + (1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)))
by Lm1
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * (((1 / 2) * (G * (i + 1),(width G))) `2 )) + (((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1)))) `2 )) + (1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)))
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * (((1 / 2) * (G * (i + 1),(width G))) `2 )) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (G * (i + 1),((width G) -' 1))) `2 ))) + (1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)))
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * ((G * (i + 1),(width G)) `2 ))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * (((1 / 2) * (G * (i + 1),((width G) -' 1))) `2 ))) + (1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)))
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * ((G * (i + 1),(width G)) `2 ))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * (i + 1),((width G) -' 1)) `2 )))) + (1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)))
by Lm3
.=
((1 / 2) * (G * (i + 1),(width G))) `2
by A16, Lm3
;
then A17: (((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|) =
|[(((1 / 2) * (G * (i + 1),(width G))) `1 ),(((1 / 2) * (G * (i + 1),(width G))) `2 )]|
by A13, EUCLID:57
.=
(1 / 2) * (G * (i + 1),(width G))
by EUCLID:57
;
(1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))) =
((1 / 2) * (G * i,(width G))) + ((1 / 2) * (G * (i + 1),(width G)))
by EUCLID:36
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) + (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 / 2) * (G * (i + 1),(width G)))
by EUCLID:33
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * i,(width G))))) + ((1 / 2) * (G * (i + 1),(width G)))
by EUCLID:37
.=
((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * i,(width G))))) + (((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1)))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|)
by A17, EUCLID:30
.=
(((((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * i,(width G)))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * i,(width G))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * (G * (i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * (G * (i + 1),((width G) -' 1))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|)
by EUCLID:30
.=
(((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + ((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * |[0 ,1]|)) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))))
by A15, EUCLID:30
.=
((1 - (1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1))) * (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|)) + ((1 / (((1 / 2) * (((G * i,(width G)) `2 ) - ((G * (i + 1),((width G) -' 1)) `2 ))) + 1)) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))))
by EUCLID:36
;
then
(1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))) in LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1))))
by A11, A12;
then A18:
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))) = (LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) \/ (LSeg ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))))
by TOPREAL1:11;
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) c= (Int (cell G,i,(width G))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}
by A1, A2, Th50;
hence
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((1 / 2) * ((G * i,(width G)) + (G * (i + 1),((width G) -' 1)))) c= ((Int (cell G,i,((width G) -' 1))) \/ (Int (cell G,i,(width G)))) \/ {((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))}
by A18, A7, A14, XBOOLE_1:13; verum