let f be V22() standard clockwise_oriented special_circular_sequence; for G being Go-board st f is_sequence_on G holds
for i, j, k being Nat st 1 <= k & k + 1 <= len f & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
(f /. k) `2 <> S-bound (L~ f)
let G be Go-board; ( f is_sequence_on G implies for i, j, k being Nat st 1 <= k & k + 1 <= len f & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
(f /. k) `2 <> S-bound (L~ f) )
assume A1:
f is_sequence_on G
; for i, j, k being Nat st 1 <= k & k + 1 <= len f & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
(f /. k) `2 <> S-bound (L~ f)
let i, j, k be Nat; ( 1 <= k & k + 1 <= len f & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) implies (f /. k) `2 <> S-bound (L~ f) )
assume that
A2:
( 1 <= k & k + 1 <= len f )
and
A3:
[i,j] in Indices G
and
A4:
[(i + 1),j] in Indices G
and
A5:
f /. k = G * (i,j)
and
A6:
f /. (k + 1) = G * ((i + 1),j)
and
A7:
(f /. k) `2 = S-bound (L~ f)
; contradiction
A8:
right_cell (f,k,G) = cell (G,i,(j -' 1))
by A1, A2, A3, A4, A5, A6, GOBRD13:24;
A9:
i <= len G
by A3, MATRIX_0:32;
A10:
j <= width G
by A3, MATRIX_0:32;
A11:
i + 1 <= len G
by A4, MATRIX_0:32;
set p = (1 / 2) * ((G * (i,(j -' 1))) + (G * ((i + 1),j)));
A12:
0 + 1 <= i
by A3, MATRIX_0:32;
A13:
1 <= j
by A3, MATRIX_0:32;
then A14:
(j -' 1) + 1 = j
by XREAL_1:235;
per cases
( j = 1 or j > 1 )
by A13, XXREAL_0:1;
suppose
j > 1
;
contradictionthen
j >= 1
+ 1
by NAT_1:13;
then A15:
j - 1
>= (1 + 1) - 1
by XREAL_1:9;
j < (width G) + 1
by A10, NAT_1:13;
then A16:
j - 1
< ((width G) + 1) - 1
by XREAL_1:9;
i < len G
by A11, NAT_1:13;
then A17:
Int (cell (G,i,(j -' 1))) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(j -' 1))) `2 < s & s < (G * (1,j)) `2 ) }
by A12, A14, A15, A16, GOBOARD6:26;
A18:
(1 / 2) * ((G * (i,(j -' 1))) + (G * ((i + 1),j))) in Int (right_cell (f,k,G))
by A12, A10, A11, A8, A14, A15, GOBOARD6:31;
then consider r,
s being
Real such that A19:
(1 / 2) * ((G * (i,(j -' 1))) + (G * ((i + 1),j))) = |[r,s]|
and
(G * (i,1)) `1 < r
and
r < (G * ((i + 1),1)) `1
and
(G * (1,(j -' 1))) `2 < s
and A20:
s < (G * (1,j)) `2
by A8, A17;
((1 / 2) * ((G * (i,(j -' 1))) + (G * ((i + 1),j)))) `2 = s
by A19, EUCLID:52;
then
((1 / 2) * ((G * (i,(j -' 1))) + (G * ((i + 1),j)))) `2 < S-bound (L~ f)
by A5, A7, A12, A9, A13, A10, A20, GOBOARD5:1;
then A21:
(1 / 2) * ((G * (i,(j -' 1))) + (G * ((i + 1),j))) in LeftComp f
by Th11;
Int (right_cell (f,k,G)) c= RightComp f
by A1, A2, JORDAN1H:25;
then
(1 / 2) * ((G * (i,(j -' 1))) + (G * ((i + 1),j))) in (LeftComp f) /\ (RightComp f)
by A18, A21, XBOOLE_0:def 4;
then
LeftComp f meets RightComp f
by XBOOLE_0:def 7;
hence
contradiction
by GOBRD14:14;
verum end; end;