let f be V22() standard clockwise_oriented special_circular_sequence; :: thesis: 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; :: thesis: ( 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 ; :: thesis: 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; :: thesis: ( 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) ; :: thesis: 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 ;
A10: ( 1 <= i + 1 & 1 <= j ) by ;
set p = (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))));
A11: ( 0 + 1 <= i & 1 <= j ) by ;
A12: j <= width G by ;
A13: i + 1 <= len G by ;
per cases ( j = width G or j < width G ) by ;
suppose j = width G ; :: thesis: contradiction
end;
suppose A14: j < width G ; :: thesis: contradiction
i < len G by ;
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 ;
j + 1 <= width G by ;
then A16: (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in Int (right_cell (f,k,G)) by ;
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 ;
((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2 = s by ;
then ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2 > N-bound (L~ f) by ;
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 ;
then (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in () /\ () by ;
then LeftComp f meets RightComp f by XBOOLE_0:def 7;
hence contradiction by GOBRD14:14; :: thesis: verum
end;
end;