let f be constant 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,(j + 1)] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
(f /. k) `1 <> W-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,(j + 1)] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
(f /. k) `1 <> W-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,(j + 1)] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
(f /. k) `1 <> W-bound (L~ f)
let i, j, k be Nat; :: thesis: ( 1 <= k & k + 1 <= len f & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) implies (f /. k) `1 <> W-bound (L~ f) )
assume that
A2: ( 1 <= k & k + 1 <= len f ) and
A3: [i,j] in Indices G and
A4: [i,(j + 1)] in Indices G and
A5: f /. k = G * (i,(j + 1)) and
A6: f /. (k + 1) = G * (i,j) and
A7: (f /. k) `1 = W-bound (L~ f) ; :: thesis: contradiction
A8: right_cell (f,k,G) = cell (G,(i -' 1),j) by A1, A2, A3, A4, A5, A6, GOBRD13:28;
A9: ( 1 <= i & i <= len G ) by A4, MATRIX_0:32;
A10: 1 <= j by A3, MATRIX_0:32;
A11: 1 <= j + 1 by A4, MATRIX_0:32;
A12: j + 1 <= width G by A4, MATRIX_0:32;
set p = (1 / 2) * ((G * ((i -' 1),j)) + (G * (i,(j + 1))));
A13: i <= len G by A3, MATRIX_0:32;
A14: 0 + 1 <= i by A3, MATRIX_0:32;
then A15: (i -' 1) + 1 = i by XREAL_1:235;

per cases ( i = 1 or i > 1 ) by A14, XXREAL_0:1;

suppose i = 1 ; :: thesis: contradiction

hence contradiction by A1, A2, A3, A4, A5, A6, Th16; :: thesis: verum

end;

suppose i > 1 ; :: thesis: contradiction

then i >= 1 + 1 by NAT_1:13;
then A16: i - 1 >= (1 + 1) - 1 by XREAL_1:9;
i < (len G) + 1 by A13, NAT_1:13;
then A17: i - 1 < ((len G) + 1) - 1 by XREAL_1:9;
j < width G by A12, NAT_1:13;
then A18: Int (cell (G,(i -' 1),j)) = { |[r,s]| where r, s is Real : ( (G * ((i -' 1),1)) `1 < r & r < (G * (i,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A10, A15, A16, A17, GOBOARD6:26;
A19: (1 / 2) * ((G * ((i -' 1),j)) + (G * (i,(j + 1)))) in Int (right_cell (f,k,G)) by A13, A10, A12, A8, A15, A16, GOBOARD6:31;
then consider r, s being Real such that
A20: (1 / 2) * ((G * ((i -' 1),j)) + (G * (i,(j + 1)))) = |[r,s]| and
(G * ((i -' 1),1)) `1 < r and
A21: r < (G * (i,1)) `1 and
(G * (1,j)) `2 < s and
s < (G * (1,(j + 1))) `2 by A8, A18;
((1 / 2) * ((G * ((i -' 1),j)) + (G * (i,(j + 1))))) `1 = r by A20, EUCLID:52;
then ((1 / 2) * ((G * ((i -' 1),j)) + (G * (i,(j + 1))))) `1 < W-bound (L~ f) by A5, A7, A9, A11, A12, A21, GOBOARD5:2;
then A22: (1 / 2) * ((G * ((i -' 1),j)) + (G * (i,(j + 1)))) in LeftComp f by Th9;
Int (right_cell (f,k,G)) c= RightComp f by A1, A2, JORDAN1H:25;
then (1 / 2) * ((G * ((i -' 1),j)) + (G * (i,(j + 1)))) in (LeftComp f) /\ (RightComp f) by A19, A22, XBOOLE_0:def 4;
then LeftComp f meets RightComp f by XBOOLE_0:def 7;
hence contradiction by GOBRD14:14; :: thesis: verum

end;

end;