let i be Element of NAT ; :: thesis: for G being Go-board st 1 <= i & i < len G holds
LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((1 / 2) * ((G * i,1) + (G * (i + 1),1))) c= (Int (cell G,i,0 )) \/ {((1 / 2) * ((G * i,1) + (G * (i + 1),1)))}
let G be Go-board; :: thesis: ( 1 <= i & i < len G implies LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((1 / 2) * ((G * i,1) + (G * (i + 1),1))) c= (Int (cell G,i,0 )) \/ {((1 / 2) * ((G * i,1) + (G * (i + 1),1)))} )
assume A1:
( 1 <= i & i < len G )
; :: thesis: LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((1 / 2) * ((G * i,1) + (G * (i + 1),1))) c= (Int (cell G,i,0 )) \/ {((1 / 2) * ((G * i,1) + (G * (i + 1),1)))}
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((1 / 2) * ((G * i,1) + (G * (i + 1),1))) or x in (Int (cell G,i,0 )) \/ {((1 / 2) * ((G * i,1) + (G * (i + 1),1)))} )
assume A2:
x in LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((1 / 2) * ((G * i,1) + (G * (i + 1),1)))
; :: thesis: x in (Int (cell G,i,0 )) \/ {((1 / 2) * ((G * i,1) + (G * (i + 1),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 * ((1 / 2) * ((G * i,1) + (G * (i + 1),1))))
and
A3:
( 0 <= r & r <= 1 )
by A2;
now per cases
( r = 1 or r < 1 )
by A3, XXREAL_0:1;
case A5:
r < 1
;
:: thesis: p in Int (cell G,i,0 )set s2 =
(G * 1,1) `2 ;
set r1 =
(G * i,1) `1 ;
set r2 =
(G * (i + 1),1) `1 ;
A6:
( 1
<= i + 1 &
i + 1
<= len G )
by A1, NAT_1:13;
0 <> width G
by GOBOARD1:def 5;
then A7:
1
<= width G
by NAT_1:14;
A8:
G * (i + 1),1 =
|[((G * (i + 1),1) `1 ),((G * (i + 1),1) `2 )]|
by EUCLID:57
.=
|[((G * (i + 1),1) `1 ),((G * 1,1) `2 )]|
by A6, A7, GOBOARD5:2
;
A9:
G * i,1 =
|[((G * i,1) `1 ),((G * i,1) `2 )]|
by EUCLID:57
.=
|[((G * i,1) `1 ),((G * 1,1) `2 )]|
by A1, A7, GOBOARD5:2
;
set r3 =
(1 - r) * (1 / 2);
set s3 =
r * (1 / 2);
A10:
p =
(((1 - r) * ((1 / 2) * ((G * i,1) + (G * (i + 1),1)))) - ((1 - r) * |[0 ,1]|)) + (r * ((1 / 2) * ((G * i,1) + (G * (i + 1),1))))
by A4, EUCLID:53
.=
((((1 - r) * (1 / 2)) * ((G * i,1) + (G * (i + 1),1))) - ((1 - r) * |[0 ,1]|)) + (r * ((1 / 2) * ((G * i,1) + (G * (i + 1),1))))
by EUCLID:34
.=
((((1 - r) * (1 / 2)) * ((G * i,1) + (G * (i + 1),1))) - |[((1 - r) * 0 ),((1 - r) * 1)]|) + (r * ((1 / 2) * ((G * i,1) + (G * (i + 1),1))))
by EUCLID:62
.=
((((1 - r) * (1 / 2)) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,(1 - r)]|) + ((r * (1 / 2)) * ((G * i,1) + (G * (i + 1),1)))
by EUCLID: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 * (1 / 2)) * ((G * i,1) + (G * (i + 1),1)))
by A8, A9, EUCLID:60
.=
((((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 * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,1) `2 ) + ((G * 1,1) `2 ))]|)
by A8, A9, EUCLID:60
.=
(|[(((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 * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,1) `2 ) + ((G * 1,1) `2 ))]|)
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 * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))),((r * (1 / 2)) * (((G * 1,1) `2 ) + ((G * 1,1) `2 )))]|
by EUCLID:62
.=
|[((((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 * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))),((r * (1 / 2)) * (((G * 1,1) `2 ) + ((G * 1,1) `2 )))]|
by EUCLID:66
.=
|[((((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 * 1,1) `2 ) + ((G * 1,1) `2 ))) - (1 - r)) + ((r * (1 / 2)) * (((G * 1,1) `2 ) + ((G * 1,1) `2 ))))]|
by EUCLID:60
;
A11:
1
- r > 0
by A5, XREAL_1:52;
then A12:
(1 - r) * (1 / 2) > (1 / 2) * 0
by XREAL_1:70;
i < i + 1
by XREAL_1:31;
then A13:
(G * i,1) `1 < (G * (i + 1),1) `1
by A1, A6, A7, GOBOARD5:4;
A14:
r * (1 / 2) >= 0
by A3;
(G * 1,1) `2 < ((G * 1,1) `2 ) + (1 - r)
by A11, XREAL_1:31;
then A15:
((G * 1,1) `2 ) - (1 - r) < (G * 1,1) `2
by XREAL_1:21;
A16:
((G * i,1) `1 ) + ((G * i,1) `1 ) < ((G * i,1) `1 ) + ((G * (i + 1),1) `1 )
by A13, XREAL_1:8;
then A17:
((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 A12, XREAL_1:70;
A18:
(r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 )) <= (r * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))
by A14, A16, XREAL_1:66;
(((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
;
then A19:
(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 A17, A18, XREAL_1:10;
A20:
((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 A21:
((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 A12, XREAL_1:70;
A22:
(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 A14, A20, XREAL_1:66;
(((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 A21, A22, XREAL_1:10;
Int (cell G,i,0 ) = { |[r',s']| where r', s' is Real : ( (G * i,1) `1 < r' & r' < (G * (i + 1),1) `1 & s' < (G * 1,1) `2 ) }
by A1, Th27;
hence
p in Int (cell G,i,0 )
by A10, A15, A19, A23;
:: thesis: verum end;
hence
x in (Int (cell G,i,0 )) \/ {((1 / 2) * ((G * i,1) + (G * (i + 1),1)))}
by XBOOLE_0:def 3; :: thesis: verum