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 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;

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 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;

end;

suppose A14:
j < width G
; :: thesis: 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; :: thesis: verum

end;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; :: thesis: verum