let G be Go-board; LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) c= (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)}
let x be object ; TARSKI:def 3 ( not x in LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) or x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} )
set r1 = (G * ((len G),1)) `1 ;
set s1 = (G * (1,(width G))) `2 ;
assume A1:
x in LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|))
; x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A2:
p = ((1 - r) * ((G * ((len G),(width G))) + |[1,1]|)) + (r * ((G * ((len G),(width G))) + |[1,0]|))
and
0 <= r
and
A3:
r <= 1
by A1;
now ( ( r = 1 & p in {((G * ((len G),(width G))) + |[1,0]|)} ) or ( r < 1 & p in Int (cell (G,(len G),(width G))) ) )per cases
( r = 1 or r < 1 )
by A3, XXREAL_0:1;
case
r < 1
;
p in Int (cell (G,(len G),(width G)))then
1
- r > 0
by XREAL_1:50;
then A4:
(G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + (1 - r)
by XREAL_1:29;
A5:
(G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + 1
by XREAL_1:29;
0 <> width G
by MATRIX_0:def 10;
then A6:
1
<= width G
by NAT_1:14;
0 <> len G
by MATRIX_0:def 10;
then A7:
1
<= len G
by NAT_1:14;
A8:
G * (
(len G),
(width G)) =
|[((G * ((len G),(width G))) `1),((G * ((len G),(width G))) `2)]|
by EUCLID:53
.=
|[((G * ((len G),1)) `1),((G * ((len G),(width G))) `2)]|
by A6, A7, GOBOARD5:2
.=
|[((G * ((len G),1)) `1),((G * (1,(width G))) `2)]|
by A6, A7, GOBOARD5:1
;
A9:
Int (cell (G,(len G),(width G))) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & (G * (1,(width G))) `2 < s9 ) }
by Th22;
p =
(((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + (r * ((G * ((len G),(width G))) + |[1,0]|))
by A2, RLVECT_1:def 5
.=
(((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + ((r * (G * ((len G),(width G)))) + (r * |[1,0]|))
by RLVECT_1:def 5
.=
((r * (G * ((len G),(width G)))) + (((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|))) + (r * |[1,0]|)
by RLVECT_1:def 3
.=
(((r * (G * ((len G),(width G)))) + ((1 - r) * (G * ((len G),(width G))))) + ((1 - r) * |[1,1]|)) + (r * |[1,0]|)
by RLVECT_1:def 3
.=
(((r + (1 - r)) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + (r * |[1,0]|)
by RLVECT_1:def 6
.=
((G * ((len G),(width G))) + ((1 - r) * |[1,1]|)) + (r * |[1,0]|)
by RLVECT_1:def 8
.=
((G * ((len G),(width G))) + |[((1 - r) * 1),((1 - r) * 1)]|) + (r * |[1,0]|)
by EUCLID:58
.=
((G * ((len G),(width G))) + |[(1 - r),(1 - r)]|) + |[(r * 1),(r * 0)]|
by EUCLID:58
.=
|[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,(width G))) `2) + (1 - r))]| + |[r,0]|
by A8, EUCLID:56
.=
|[((((G * ((len G),1)) `1) + (1 - r)) + r),((((G * (1,(width G))) `2) + (1 - r)) + 0)]|
by EUCLID:56
.=
|[(((G * ((len G),1)) `1) + 1),(((G * (1,(width G))) `2) + (1 - r))]|
;
hence
p in Int (cell (G,(len G),(width G)))
by A4, A5, A9;
verum end; end; end;
hence
x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)}
by XBOOLE_0:def 3; verum