let f be non 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 Element of 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
j < width G
let G be Go-board; :: thesis: ( f is_sequence_on G implies for i, j, k being Element of 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
j < width G )
assume A1: f is_sequence_on G ; :: thesis: for i, j, k being Element of 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
j < width G
A2: L~ f is Bounded by JORDAN2C:73;
let i, j, k be Element of 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 j < width G )
assume that
A3: ( 1 <= k & k + 1 <= len f ) and
A4: [i,j] in Indices G and
A5: ( [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) ) ; :: thesis: j < width G
assume A6: j >= width G ; :: thesis: contradiction
j <= width G by A4, MATRIX_1:39;
then A7: j = width G by A6, XXREAL_0:1;
A8: i <= len G by A4, MATRIX_1:39;
right_cell (f,k,G) = cell (G,i,j) by A1, A3, A4, A5, GOBRD13:27;
then not (right_cell (f,k,G)) \ (L~ f) is Bounded by A8, A7, A2, JORDAN1A:48, TOPREAL6:99;
then not RightComp f is Bounded by A1, A3, JORDAN2C:16, JORDAN9:29;
then not BDD (L~ f) is Bounded by GOBRD14:47;
hence contradiction by JORDAN2C:73, JORDAN2C:114; :: thesis: verum