let G be Go-board; ( 1 < len G & 1 < width G implies LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)} )
assume that
A1:
1 < len G
and
A2:
1 < width G
; LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)}
set q2 = G * (1,1);
set q3 = G * (1,2);
set r = 1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1);
A3:
0 + (1 + 1) <= width G
by A2, NAT_1:13;
then A4:
(G * (1,1)) `1 = (G * (1,2)) `1
by A1, GOBOARD5:2;
(G * (1,1)) `2 < (G * (1,2)) `2
by A1, A3, GOBOARD5:4;
then A5:
((G * (1,2)) `2) - ((G * (1,1)) `2) > 0
by XREAL_1:50;
then
1 < ((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1
by XREAL_1:29, XREAL_1:129;
then A6:
1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1) < 1
by XREAL_1:212;
A7: (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|))) `1 =
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) `1) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `1)
by Lm1
.=
((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) - |[1,1]|) `1)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `1)
by Lm3
.=
((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) - |[1,1]|) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|) `1))
by Lm3
.=
((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) `1) - (|[1,1]| `1))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|) `1))
by Lm2
.=
((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) `1) - (|[1,1]| `1))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1) - (|[1,0]| `1)))
by Lm2
.=
((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) `1) - 1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1) - (|[1,0]| `1)))
by EUCLID:52
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1) - 1))
by EUCLID:52
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1))) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) + (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) + (G * (1,2))) `1)))) - 1
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) `1) + ((G * (1,1)) `1))))) - 1
by A4, Lm1
.=
((G * (1,1)) `1) - (|[1,0]| `1)
by EUCLID:52
.=
((G * (1,1)) - |[1,0]|) `1
by Lm2
;
A8: (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * ((G * (1,2)) `2))) - ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * ((G * (1,1)) `2)))) + (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) =
(1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)
.=
1
by A5, XCMPLX_1:106
;
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|))) `2 =
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2)
by Lm1
.=
((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2)
by RLVECT_1:34
.=
((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) `2) - (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * |[1,1]|) `2)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2)
by Lm2
.=
((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) `2) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (|[1,1]| `2))) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2)
by Lm3
.=
((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) `2) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2)
by EUCLID:52
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2)
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|) `2))
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `2) - (|[1,0]| `2)))
by Lm2
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `2) - 0))
by EUCLID:52
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) + (G * (1,2))) `2)))
by Lm3
.=
(((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) `2) + ((G * (1,2)) `2))))
by Lm1
.=
((G * (1,1)) `2) - 0
by A8
.=
((G * (1,1)) `2) - (|[1,0]| `2)
by EUCLID:52
.=
((G * (1,1)) - |[1,0]|) `2
by Lm2
;
then ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) =
|[(((G * (1,1)) - |[1,0]|) `1),(((G * (1,1)) - |[1,0]|) `2)]|
by A7, EUCLID:53
.=
(G * (1,1)) - |[1,0]|
by EUCLID:53
;
then
(G * (1,1)) - |[1,0]| in LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|))
by A5, A6;
then A9:
LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) = (LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|))) \/ (LSeg (((G * (1,1)) - |[1,0]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)))
by TOPREAL1:5;
set I1 = Int (cell (G,0,0));
set I2 = Int (cell (G,0,1));
(0 + 1) + 1 = 0 + (1 + 1)
;
then A10:
LSeg (((G * (1,1)) - |[1,0]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= (Int (cell (G,0,1))) \/ {((G * (1,1)) - |[1,0]|)}
by A2, Th48;
A11: ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)} =
(Int (cell (G,0,0))) \/ ((Int (cell (G,0,1))) \/ ({((G * (1,1)) - |[1,0]|)} \/ {((G * (1,1)) - |[1,0]|)}))
by XBOOLE_1:4
.=
(Int (cell (G,0,0))) \/ (((Int (cell (G,0,1))) \/ {((G * (1,1)) - |[1,0]|)}) \/ {((G * (1,1)) - |[1,0]|)})
by XBOOLE_1:4
.=
((Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)}) \/ ((Int (cell (G,0,1))) \/ {((G * (1,1)) - |[1,0]|)})
by XBOOLE_1:4
;
LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)}
by Th56;
hence
LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)}
by A9, A10, A11, XBOOLE_1:13; verum