let i be Nat; for G being Go-board st 1 <= i & i < len G holds
LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)}
let G be Go-board; ( 1 <= i & i < len G implies LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} )
assume that
A1:
1 <= i
and
A2:
i < len G
; LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)}
let x be object ; TARSKI:def 3 ( not x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) or x in (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} )
assume A3:
x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|))
; x in (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A4:
p = ((1 - r) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|)) + (r * ((G * ((i + 1),1)) - |[0,1]|))
and
A5:
0 <= r
and
A6:
r <= 1
by A3;
now ( ( r = 1 & p in {((G * ((i + 1),1)) - |[0,1]|)} ) or ( r < 1 & p in Int (cell (G,i,0)) ) )per cases
( r = 1 or r < 1 )
by A6, XXREAL_0:1;
case A7:
r < 1
;
p in Int (cell (G,i,0))set r3 =
(1 - r) * (1 / 2);
1
- r > 0
by A7, XREAL_1:50;
then A8:
(1 - r) * (1 / 2) > (1 / 2) * 0
by XREAL_1:68;
set s1 =
(G * (1,1)) `2 ;
set r1 =
(G * (i,1)) `1 ;
set r2 =
(G * ((i + 1),1)) `1 ;
A9:
(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + (r * ((G * (i,1)) `1)) = (G * (i,1)) `1
;
A10:
i + 1
<= len G
by A2, NAT_1:13;
0 <> width G
by MATRIX_0:def 10;
then A11:
1
<= width G
by NAT_1:14;
i < i + 1
by XREAL_1:29;
then A12:
(G * (i,1)) `1 < (G * ((i + 1),1)) `1
by A1, A10, A11, GOBOARD5:3;
then
((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)
by XREAL_1:6;
then A13:
((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))
by A8, XREAL_1:68;
r * ((G * (i,1)) `1) <= r * ((G * ((i + 1),1)) `1)
by A5, A12, XREAL_1:64;
then A14:
(G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1))
by A13, A9, XREAL_1:8;
A15:
1
<= i + 1
by A1, NAT_1:13;
A16:
G * (
i,1) =
|[((G * (i,1)) `1),((G * (i,1)) `2)]|
by EUCLID:53
.=
|[((G * (i,1)) `1),((G * (1,1)) `2)]|
by A1, A2, A11, GOBOARD5:1
;
(G * (1,1)) `2 < ((G * (1,1)) `2) + 1
by XREAL_1:29;
then A17:
((G * (1,1)) `2) - 1
< (G * (1,1)) `2
by XREAL_1:19;
A18:
(((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) = (G * ((i + 1),1)) `1
;
((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)
by A12, XREAL_1:6;
then
((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))
by A8, XREAL_1:68;
then A19:
(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) < (G * ((i + 1),1)) `1
by A18, XREAL_1:8;
A20:
Int (cell (G,i,0)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & s9 < (G * (1,1)) `2 ) }
by A1, A2, Th24;
A21:
G * (
(i + 1),1) =
|[((G * ((i + 1),1)) `1),((G * ((i + 1),1)) `2)]|
by EUCLID:53
.=
|[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]|
by A15, A10, A11, GOBOARD5:1
;
p =
(((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[0,1]|)) + (r * ((G * ((i + 1),1)) - |[0,1]|))
by A4, RLVECT_1:34
.=
((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[0,1]|)) + (r * ((G * ((i + 1),1)) - |[0,1]|))
by RLVECT_1:def 7
.=
((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((G * ((i + 1),1)) - |[0,1]|))
by EUCLID:58
.=
((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (r * (|[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]| - |[0,1]|))
by A21, A16, EUCLID:56
.=
((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + ((r * |[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]|) - (r * |[0,1]|))
by RLVECT_1:34
.=
((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (|[(r * ((G * ((i + 1),1)) `1)),(r * ((G * (1,1)) `2))]| - (r * |[0,1]|))
by EUCLID:58
.=
((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (|[(r * ((G * ((i + 1),1)) `1)),(r * ((G * (1,1)) `2))]| - |[(r * 0),(r * 1)]|)
by EUCLID:58
.=
((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + |[((r * ((G * ((i + 1),1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]|
by EUCLID:62
.=
(|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| - |[0,(1 - r)]|) + |[((r * ((G * ((i + 1),1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]|
by EUCLID:58
.=
|[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) - 0),((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r))]| + |[((r * ((G * ((i + 1),1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]|
by EUCLID:62
.=
|[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1))),(((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r)) + ((r * ((G * (1,1)) `2)) - r))]|
by EUCLID:56
;
hence
p in Int (cell (G,i,0))
by A17, A14, A19, A20;
verum end;
hence
x in (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)}
by XBOOLE_0:def 3; verum