let i, j be Element of NAT ; :: thesis: 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; :: thesis: ( 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 A1:
( 1 <= i & i < len G & 1 <= j & j < width G )
; :: thesis: 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 set ; :: according to TARSKI:def 3 :: thesis: ( 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 A2:
x in LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))
; :: thesis: 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
A3:
( 0 <= r & r <= 1 )
and
A4:
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 A2, SPPOL_1:21;
now per cases
( r = 1 or r < 1 )
by A3, XXREAL_0:1;
case A5:
r < 1
;
:: thesis: p in Int (cell G,i,j)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 ;
A6:
( 1
<= j + 1 &
j + 1
<= width G )
by A1, NAT_1:13;
A7:
( 1
<= i + 1 &
i + 1
<= len G )
by A1, NAT_1:13;
A8:
G * i,
j =
|[((G * i,j) `1 ),((G * i,j) `2 )]|
by EUCLID:57
.=
|[((G * i,1) `1 ),((G * i,j) `2 )]|
by A1, GOBOARD5:3
.=
|[((G * i,1) `1 ),((G * 1,j) `2 )]|
by A1, GOBOARD5:2
;
A9:
G * (i + 1),
(j + 1) =
|[((G * (i + 1),(j + 1)) `1 ),((G * (i + 1),(j + 1)) `2 )]|
by EUCLID:57
.=
|[((G * (i + 1),1) `1 ),((G * (i + 1),(j + 1)) `2 )]|
by A6, A7, GOBOARD5:3
.=
|[((G * (i + 1),1) `1 ),((G * 1,(j + 1)) `2 )]|
by A6, A7, GOBOARD5:2
;
A10:
G * i,
(j + 1) =
|[((G * i,(j + 1)) `1 ),((G * i,(j + 1)) `2 )]|
by EUCLID:57
.=
|[((G * i,1) `1 ),((G * i,(j + 1)) `2 )]|
by A1, A6, GOBOARD5:3
.=
|[((G * i,1) `1 ),((G * 1,(j + 1)) `2 )]|
by A1, A6, GOBOARD5:2
;
set r3 =
(1 - r) * (1 / 2);
set s3 =
r * (1 / 2);
A11:
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 A4, EUCLID:34
.=
(((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 EUCLID:34
.=
(((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 A8, A9, EUCLID:60
.=
(((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 A9, A10, EUCLID:60
.=
|[(((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:62
.=
|[(((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:62
.=
|[((((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:60
;
1
- r > 0
by A5, XREAL_1:52;
then A12:
(1 - r) * (1 / 2) > (1 / 2) * 0
by XREAL_1:70;
0 <> width G
by GOBOARD1:def 5;
then A13:
1
<= width G
by NAT_1:14;
i < i + 1
by XREAL_1:31;
then A14:
(G * i,1) `1 < (G * (i + 1),1) `1
by A1, A7, A13, GOBOARD5:4;
then A15:
((G * i,1) `1 ) + ((G * i,1) `1 ) < ((G * i,1) `1 ) + ((G * (i + 1),1) `1 )
by XREAL_1:8;
then A16:
((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;
0 <> len G
by GOBOARD1:def 5;
then A17:
1
<= len G
by NAT_1:14;
j < j + 1
by XREAL_1:31;
then A18:
(G * 1,j) `2 < (G * 1,(j + 1)) `2
by A1, A6, A17, GOBOARD5:5;
A19:
r * (1 / 2) >= 0
by A3;
then A20:
(r * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 )) <= (r * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))
by A15, 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 A21:
(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 A16, A20, XREAL_1:10;
((G * i,1) `1 ) + ((G * (i + 1),1) `1 ) < ((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 )
by A14, XREAL_1:8;
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 A12, XREAL_1:70;
((G * i,1) `1 ) + ((G * (i + 1),1) `1 ) < ((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 )
by A14, XREAL_1:10;
then A23:
(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 A19, 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 A24:
(((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, A23, XREAL_1:10;
A25:
((G * 1,j) `2 ) + ((G * 1,j) `2 ) < ((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 )
by A18, XREAL_1:8;
then A26:
((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 A12, XREAL_1:70;
((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) < ((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 )
by A18, XREAL_1:8;
then
((G * 1,j) `2 ) + ((G * 1,j) `2 ) < ((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 )
by A25, XXREAL_0:2;
then A27:
(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 A19, XREAL_1:66;
(((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
;
then A28:
(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 A26, A27, XREAL_1:10;
((G * 1,j) `2 ) + ((G * 1,(j + 1)) `2 ) < ((G * 1,(j + 1)) `2 ) + ((G * 1,(j + 1)) `2 )
by A18, XREAL_1:8;
then A29:
((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 A12, XREAL_1:70;
(((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
;
then A30:
(((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 A29, XREAL_1:10;
Int (cell G,i,j) = { |[r',s']| where r', s' is Real : ( (G * i,1) `1 < r' & r' < (G * (i + 1),1) `1 & (G * 1,j) `2 < s' & s' < (G * 1,(j + 1)) `2 ) }
by A1, Th29;
hence
p in Int (cell G,i,j)
by A11, A21, A24, A28, A30;
:: thesis: 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; :: thesis: verum