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,j) & f /. (k + 1) = G * ((i + 1),j) holds

(f /. k) `2 <> S-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,j) & f /. (k + 1) = G * ((i + 1),j) holds

(f /. k) `2 <> S-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,j) & f /. (k + 1) = G * ((i + 1),j) holds

(f /. k) `2 <> S-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,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) ; :: thesis: 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;

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; :: 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,j) & f /. (k + 1) = G * ((i + 1),j) holds

(f /. k) `2 <> S-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,j) & f /. (k + 1) = G * ((i + 1),j) holds

(f /. k) `2 <> S-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,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) ; :: thesis: 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;

end;

suppose
j > 1
; :: thesis: contradiction

then
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; :: thesis: verum

end;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; :: thesis: verum