let j be Nat; for G being Go-board st 1 <= j & j + 1 <= width G holds
((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j))
let G be Go-board; ( 1 <= j & j + 1 <= width G implies ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j)) )
assume that
A1:
1 <= j
and
A2:
j + 1 <= width G
; ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j))
set s1 = (G * (1,j)) `2 ;
set r1 = (G * (1,j)) `1 ;
set s2 = (G * (1,(j + 1))) `2 ;
len G <> 0
by MATRIX_0:def 10;
then A3:
1 <= len G
by NAT_1:14;
len G <> 0
by MATRIX_0:def 10;
then A4:
1 <= len G
by NAT_1:14;
j < j + 1
by XREAL_1:29;
then A5:
(G * (1,j)) `2 < (G * (1,(j + 1))) `2
by A1, A2, A4, GOBOARD5:4;
then
((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)
by XREAL_1:6;
then A6:
(1 / 2) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < (1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))
by XREAL_1:68;
j < width G
by A2, NAT_1:13;
then A7:
(G * (1,1)) `1 = (G * (1,j)) `1
by A1, A3, GOBOARD5:2;
then
(G * (1,j)) `1 < ((G * (1,1)) `1) + 1
by XREAL_1:29;
then A8:
((G * (1,j)) `1) - 1 < (G * (1,1)) `1
by XREAL_1:19;
1 <= j + 1
by NAT_1:11;
then
(G * (1,1)) `1 = (G * (1,(j + 1))) `1
by A2, A3, GOBOARD5:2;
then
( G * (1,j) = |[((G * (1,j)) `1),((G * (1,j)) `2)]| & G * (1,(j + 1)) = |[((G * (1,j)) `1),((G * (1,(j + 1))) `2)]| )
by A7, EUCLID:53;
then
( (1 / 2) * (((G * (1,j)) `1) + ((G * (1,j)) `1)) = (G * (1,j)) `1 & (G * (1,j)) + (G * (1,(j + 1))) = |[(((G * (1,j)) `1) + ((G * (1,j)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]| )
by EUCLID:56;
then
(1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))) = |[((G * (1,j)) `1),((1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]|
by EUCLID:58;
then A9: ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| =
|[(((G * (1,j)) `1) - 1),(((1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) - 0)]|
by EUCLID:62
.=
|[(((G * (1,j)) `1) - 1),((1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]|
;
((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)
by A5, XREAL_1:6;
then A10:
(1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < (1 / 2) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))
by XREAL_1:68;
j < width G
by A2, NAT_1:13;
then
Int (cell (G,0,j)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) }
by A1, Th20;
hence
((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j))
by A9, A6, A10, A8; verum