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 + 1),j) & f /. (k + 1) = G * (i,j) holds
(f /. k) `2 <> N-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 + 1),j) & f /. (k + 1) = G * (i,j) holds
(f /. k) `2 <> N-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 + 1),j) & f /. (k + 1) = G * (i,j) holds
(f /. k) `2 <> N-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 + 1),j) & f /. (k + 1) = G * (i,j) implies (f /. k) `2 <> N-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 + 1),j)
and
A6:
f /. (k + 1) = G * (i,j)
and
A7:
(f /. k) `2 = N-bound (L~ f)
; contradiction
A8:
right_cell (f,k,G) = cell (G,i,j)
by A1, A2, A3, A4, A5, A6, GOBRD13:26;
A9:
j <= width G
by A4, MATRIX_0:32;
A10:
( 1 <= i + 1 & 1 <= j )
by A4, MATRIX_0:32;
set p = (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))));
A11:
( 0 + 1 <= i & 1 <= j )
by A3, MATRIX_0:32;
A12:
j <= width G
by A3, MATRIX_0:32;
A13:
i + 1 <= len G
by A4, MATRIX_0:32;
per cases
( j = width G or j < width G )
by A12, XXREAL_0:1;
suppose A14:
j < width G
;
contradiction
i < len G
by A13, NAT_1:13;
then A15:
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 A11, A14, GOBOARD6:26;
j + 1
<= width G
by A14, NAT_1:13;
then A16:
(1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in Int (right_cell (f,k,G))
by A11, A13, A8, GOBOARD6:31;
then consider r,
s being
Real such that A17:
(1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = |[r,s]|
and
(G * (i,1)) `1 < r
and
r < (G * ((i + 1),1)) `1
and A18:
(G * (1,j)) `2 < s
and
s < (G * (1,(j + 1))) `2
by A8, A15;
((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2 = s
by A17, EUCLID:52;
then
((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2 > N-bound (L~ f)
by A5, A7, A13, A10, A9, A18, GOBOARD5:1;
then A19:
(1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in LeftComp f
by Th12;
Int (right_cell (f,k,G)) c= RightComp f
by A1, A2, JORDAN1H:25;
then
(1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in (LeftComp f) /\ (RightComp f)
by A16, A19, XBOOLE_0:def 4;
then
LeftComp f meets RightComp f
by XBOOLE_0:def 7;
hence
contradiction
by GOBRD14:14;
verum end; end;