let i be Element of NAT ; :: thesis: for G being Go-board st 1 < width G & 1 <= i & i + 1 < len G holds
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),(((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|) c= ((Int (cell G,i,(width G))) \/ (Int (cell G,(i + 1),(width G)))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}
let G be Go-board; :: thesis: ( 1 < width G & 1 <= i & i + 1 < len G implies LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),(((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|) c= ((Int (cell G,i,(width G))) \/ (Int (cell G,(i + 1),(width G)))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)} )
assume A1:
( 1 < width G & 1 <= i & i + 1 < len G )
; :: thesis: LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),(((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|) c= ((Int (cell G,i,(width G))) \/ (Int (cell G,(i + 1),(width G)))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}
set p1 = G * i,(width G);
set p2 = G * (i + 1),(width G);
set q3 = G * (i + 2),(width G);
set r = (((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ));
A2:
i + 1 >= 1
by NAT_1:11;
i <= i + 1
by NAT_1:11;
then A3:
i < len G
by A1, XXREAL_0:2;
then A4: (G * i,(width G)) `2 =
(G * 1,(width G)) `2
by A1, GOBOARD5:2
.=
(G * (i + 1),(width G)) `2
by A1, A2, GOBOARD5:2
;
A5:
(i + 1) + 1 = i + (1 + 1)
;
then A6:
i + (1 + 1) <= len G
by A1, NAT_1:13;
A7:
i + 2 >= 1
by A5, NAT_1:11;
A8: (G * (i + 1),(width G)) `2 =
(G * 1,(width G)) `2
by A1, A2, GOBOARD5:2
.=
(G * (i + 2),(width G)) `2
by A1, A6, A7, GOBOARD5:2
;
i + 1 < i + 2
by XREAL_1:8;
then
(G * (i + 1),(width G)) `1 < (G * (i + 2),(width G)) `1
by A1, A2, A6, GOBOARD5:4;
then A9:
((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 ) < ((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )
by XREAL_1:11;
i < i + 1
by XREAL_1:31;
then
(G * i,(width G)) `1 < (G * (i + 1),(width G)) `1
by A1, GOBOARD5:4;
then A10:
((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 ) > 0
by XREAL_1:52;
then A11:
0 < (((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))
by A9, XREAL_1:141;
A12:
(((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )) < 1
by A9, A10, XREAL_1:191;
((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )) = ((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )
by A9, A10, XCMPLX_1:88;
then A13:
(G * (i + 1),(width G)) `1 = ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((G * i,(width G)) `1 )) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((G * (i + 2),(width G)) `1 ))
;
A14: 1 * ((G * (i + 1),(width G)) `2 ) =
((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((G * i,(width G)) `2 )) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((G * (i + 2),(width G)) `2 ))
by A4, A8
.=
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) `2 ) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((G * (i + 2),(width G)) `2 ))
by Lm3
.=
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) `2 ) + ((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))) `2 )
by Lm3
.=
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))) `2
by Lm1
;
1 * ((G * (i + 1),(width G)) `1 ) =
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) `1 ) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((G * (i + 2),(width G)) `1 ))
by A13, Lm3
.=
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) `1 ) + ((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))) `1 )
by Lm3
.=
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))) `1
by Lm1
;
then A15: ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))) =
|[((G * (i + 1),(width G)) `1 ),((G * (i + 1),(width G)) `2 )]|
by A14, EUCLID:57
.=
G * (i + 1),(width G)
by EUCLID:57
;
A16:
now G * (i + 1),
(width G) =
1
* (G * (i + 1),(width G))
by EUCLID:33
.=
((1 / 2) * (G * (i + 1),(width G))) + ((1 / 2) * (G * (i + 1),(width G)))
by Lm6, EUCLID:37
.=
((1 / 2) * (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) + ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G)))) + ((1 / 2) * (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))))
by A15, EUCLID:33
.=
((1 / 2) * (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))))) + ((1 / 2) * (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))))
by EUCLID:37
.=
(((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))))) + ((1 / 2) * (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))))
by EUCLID:36
.=
(((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))))) + (((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))))
by EUCLID:36
.=
((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G)))) + (((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G)))) + (((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))))))
by EUCLID:30
.=
((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G)))) + (((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G)))) + (((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))))))
by EUCLID:30
.=
(((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G)))) + ((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))))) + (((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))))
by EUCLID:30
.=
((((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G)))) + ((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))))
by EUCLID:30
.=
(((1 / 2) * (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * (i + 1),(width G))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (G * i,(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))))
by EUCLID:36
.=
(((1 / 2) * ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))))
by EUCLID:36
.=
((((1 / 2) * (1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))))) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G))))
by EUCLID:34
;
hence G * (i + 1),
(width G) =
(((1 / 2) * (1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))))) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + (((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))))
by EUCLID:30
.=
(((1 / 2) * (1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))))) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + ((1 / 2) * ((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 1),(width G))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (G * (i + 2),(width G)))))
by EUCLID:36
.=
(((1 / 2) * (1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))))) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))))
by EUCLID:36
.=
((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + ((1 / 2) * (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))))
by EUCLID:34
.=
((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + (((1 / 2) * ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G))))
by EUCLID:34
.=
((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))))
by EUCLID:34
;
:: thesis: verum end;
((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|)) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|)) =
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * |[0 ,1]|)) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * (((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|))
by EUCLID:36
.=
(((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * |[0 ,1]|)) + ((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G))))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * |[0 ,1]|))
by EUCLID:36
.=
((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G))))) + (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * |[0 ,1]|))) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * |[0 ,1]|)
by EUCLID:30
.=
(((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G))))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * |[0 ,1]|)) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * |[0 ,1]|)
by EUCLID:30
.=
((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G))))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))))) + (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * |[0 ,1]|) + (((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * |[0 ,1]|))
by EUCLID:30
.=
((((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 ))) * ((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G))))) + ((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))))) + (((1 - ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) + ((((G * (i + 1),(width G)) `1 ) - ((G * i,(width G)) `1 )) / (((G * (i + 2),(width G)) `1 ) - ((G * i,(width G)) `1 )))) * |[0 ,1]|)
by EUCLID:37
.=
(G * (i + 1),(width G)) + |[0 ,1]|
by A16, EUCLID:33
;
then
(G * (i + 1),(width G)) + |[0 ,1]| in LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),(((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|)
by A11, A12;
then A17:
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),(((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|) = (LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((G * (i + 1),(width G)) + |[0 ,1]|)) \/ (LSeg ((G * (i + 1),(width G)) + |[0 ,1]|),(((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|))
by A11, A12, TOPREAL1:11;
set I1 = Int (cell G,i,(width G));
set I2 = Int (cell G,(i + 1),(width G));
A18:
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),((G * (i + 1),(width G)) + |[0 ,1]|) c= (Int (cell G,i,(width G))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}
by A1, A3, Th58;
A19:
LSeg (((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|),((G * (i + 1),(width G)) + |[0 ,1]|) c= (Int (cell G,(i + 1),(width G))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}
by A1, A5, Th57, NAT_1:11;
((Int (cell G,i,(width G))) \/ (Int (cell G,(i + 1),(width G)))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)} =
(Int (cell G,i,(width G))) \/ ((Int (cell G,(i + 1),(width G))) \/ ({((G * (i + 1),(width G)) + |[0 ,1]|)} \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}))
by XBOOLE_1:4
.=
(Int (cell G,i,(width G))) \/ (((Int (cell G,(i + 1),(width G))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)})
by XBOOLE_1:4
.=
((Int (cell G,i,(width G))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}) \/ ((Int (cell G,(i + 1),(width G))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)})
by XBOOLE_1:4
;
hence
LSeg (((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]|),(((1 / 2) * ((G * (i + 1),(width G)) + (G * (i + 2),(width G)))) + |[0 ,1]|) c= ((Int (cell G,i,(width G))) \/ (Int (cell G,(i + 1),(width G)))) \/ {((G * (i + 1),(width G)) + |[0 ,1]|)}
by A17, A18, A19, XBOOLE_1:13; :: thesis: verum