let G be Go-board; LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)}
let x be object ; TARSKI:def 3 ( not x in LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|)) or x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)} )
set r1 = (G * (1,1)) `1 ;
set s1 = (G * (1,1)) `2 ;
assume A1:
x in LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|))
; x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A2:
p = ((1 - r) * ((G * (1,1)) - |[1,1]|)) + (r * ((G * (1,1)) - |[0,1]|))
and
0 <= r
and
A3:
r <= 1
by A1;
now ( ( r = 1 & p in {((G * (1,1)) - |[0,1]|)} ) or ( r < 1 & p in Int (cell (G,0,0)) ) )per cases
( r = 1 or r < 1 )
by A3, XXREAL_0:1;
case
r < 1
;
p in Int (cell (G,0,0))then
1
- r > 0
by XREAL_1:50;
then
(G * (1,1)) `1 < ((G * (1,1)) `1) + (1 - r)
by XREAL_1:29;
then A4:
((G * (1,1)) `1) - (1 - r) < (G * (1,1)) `1
by XREAL_1:19;
A5:
G * (1,1)
= |[((G * (1,1)) `1),((G * (1,1)) `2)]|
by EUCLID:53;
(G * (1,1)) `2 < ((G * (1,1)) `2) + 1
by XREAL_1:29;
then A6:
((G * (1,1)) `2) - 1
< (G * (1,1)) `2
by XREAL_1:19;
A7:
Int (cell (G,0,0)) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & s9 < (G * (1,1)) `2 ) }
by Th18;
p =
(((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|)) + (r * ((G * (1,1)) - |[0,1]|))
by A2, RLVECT_1:34
.=
(((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|)) + ((r * (G * (1,1))) - (r * |[0,1]|))
by RLVECT_1:34
.=
((r * (G * (1,1))) + (((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|))) - (r * |[0,1]|)
by RLVECT_1:def 3
.=
(((r * (G * (1,1))) + ((1 - r) * (G * (1,1)))) - ((1 - r) * |[1,1]|)) - (r * |[0,1]|)
by RLVECT_1:def 3
.=
(((r + (1 - r)) * (G * (1,1))) - ((1 - r) * |[1,1]|)) - (r * |[0,1]|)
by RLVECT_1:def 6
.=
((G * (1,1)) - ((1 - r) * |[1,1]|)) - (r * |[0,1]|)
by RLVECT_1:def 8
.=
((G * (1,1)) - |[((1 - r) * 1),((1 - r) * 1)]|) - (r * |[0,1]|)
by EUCLID:58
.=
((G * (1,1)) - |[(1 - r),(1 - r)]|) - |[(r * 0),(r * 1)]|
by EUCLID:58
.=
|[(((G * (1,1)) `1) - (1 - r)),(((G * (1,1)) `2) - (1 - r))]| - |[0,r]|
by A5, EUCLID:62
.=
|[((((G * (1,1)) `1) - (1 - r)) - 0),((((G * (1,1)) `2) - (1 - r)) - r)]|
by EUCLID:62
.=
|[(((G * (1,1)) `1) - (1 - r)),(((G * (1,1)) `2) - 1)]|
;
hence
p in Int (cell (G,0,0))
by A6, A4, A7;
verum end; end; end;
hence
x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)}
by XBOOLE_0:def 3; verum