let i, j be Nat; for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds
LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}
let G be Go-board; ( 1 <= i & i < len G & 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} )
assume that
A1:
1 <= i
and
A2:
i < len G
and
A3:
1 <= j
and
A4:
j < width G
; LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}
let x be object ; TARSKI:def 3 ( not x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) or x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} )
assume A5:
x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))
; x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A6:
p = ((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))
and
A7:
0 <= r
and
A8:
r <= 1
by A5;
now ( ( r = 1 & p in {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} ) or ( r < 1 & p in Int (cell (G,i,j)) ) )per cases
( r = 1 or r < 1 )
by A8, XXREAL_0:1;
case A9:
r < 1
;
p in Int (cell (G,i,j))set r3 =
(1 - r) * (1 / 2);
set s3 =
r * (1 / 2);
set r1 =
(G * (i,1)) `1 ;
set r2 =
(G * ((i + 1),1)) `1 ;
set s1 =
(G * (1,j)) `2 ;
set s2 =
(G * (1,(j + 1))) `2 ;
A10:
(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1
;
0 <> width G
by MATRIX_0:def 10;
then A11:
1
<= width G
by NAT_1:14;
A12:
i + 1
<= len G
by A2, NAT_1:13;
i < i + 1
by XREAL_1:29;
then A13:
(G * (i,1)) `1 < (G * ((i + 1),1)) `1
by A1, A12, A11, GOBOARD5:3;
then A14:
((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)
by XREAL_1:6;
then A15:
(r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))
by A7, XREAL_1:64;
1
- r > 0
by A9, XREAL_1:50;
then A16:
(1 - r) * (1 / 2) > (1 / 2) * 0
by XREAL_1:68;
then
((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 A14, XREAL_1:68;
then A17:
(G * (i,1)) `1 < (((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 A15, A10, XREAL_1:8;
0 <> len G
by MATRIX_0:def 10;
then A18:
1
<= len G
by NAT_1:14;
A19:
1
<= i + 1
by A1, NAT_1:13;
((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)
by A13, XREAL_1:8;
then A20:
(r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))
by A7, XREAL_1:64;
A21:
j + 1
<= width G
by A4, NAT_1:13;
((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)
by A13, XREAL_1:6;
then A22:
((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 A16, XREAL_1:68;
(((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1
;
then A23:
(((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))) < (G * ((i + 1),1)) `1
by A22, A20, XREAL_1:8;
A24:
Int (cell (G,i,j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) }
by A1, A2, A3, A4, Th26;
A25:
1
<= j + 1
by A3, NAT_1:13;
j < j + 1
by XREAL_1:29;
then A26:
(G * (1,j)) `2 < (G * (1,(j + 1))) `2
by A3, A21, A18, GOBOARD5:4;
then A27:
((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)
by XREAL_1:6;
A28:
G * (
(i + 1),
(j + 1)) =
|[((G * ((i + 1),(j + 1))) `1),((G * ((i + 1),(j + 1))) `2)]|
by EUCLID:53
.=
|[((G * ((i + 1),1)) `1),((G * ((i + 1),(j + 1))) `2)]|
by A25, A21, A19, A12, GOBOARD5:2
.=
|[((G * ((i + 1),1)) `1),((G * (1,(j + 1))) `2)]|
by A25, A21, A19, A12, GOBOARD5:1
;
((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)
by A26, XREAL_1:6;
then
((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)
by A27, XXREAL_0:2;
then A29:
(r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))
by A7, XREAL_1:64;
A30:
G * (
i,
j) =
|[((G * (i,j)) `1),((G * (i,j)) `2)]|
by EUCLID:53
.=
|[((G * (i,1)) `1),((G * (i,j)) `2)]|
by A1, A2, A3, A4, GOBOARD5:2
.=
|[((G * (i,1)) `1),((G * (1,j)) `2)]|
by A1, A2, A3, A4, GOBOARD5:1
;
A31:
(((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2
;
A32:
G * (
i,
(j + 1)) =
|[((G * (i,(j + 1))) `1),((G * (i,(j + 1))) `2)]|
by EUCLID:53
.=
|[((G * (i,1)) `1),((G * (i,(j + 1))) `2)]|
by A1, A2, A25, A21, GOBOARD5:2
.=
|[((G * (i,1)) `1),((G * (1,(j + 1))) `2)]|
by A1, A2, A25, A21, GOBOARD5:1
;
A33:
(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2
;
((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)
by A26, XREAL_1:6;
then
((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))
by A16, XREAL_1:68;
then A34:
(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) < (G * (1,(j + 1))) `2
by A31, XREAL_1:8;
((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))
by A16, A27, XREAL_1:68;
then A35:
(G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)))
by A29, A33, XREAL_1:8;
p =
(((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))
by A6, RLVECT_1:def 7
.=
(((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + ((r * (1 / 2)) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))
by RLVECT_1:def 7
.=
(((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))
by A30, A28, EUCLID:56
.=
(((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))]|)
by A28, A32, EUCLID:56
.=
|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))]|)
by EUCLID:58
.=
|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)))]|
by EUCLID:58
.=
|[((((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)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))))]|
by EUCLID:56
;
hence
p in Int (cell (G,i,j))
by A17, A23, A35, A34, A24;
verum end;
hence
x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}
by XBOOLE_0:def 3; verum