let n, i be Nat; :: thesis: for G being Go-board
for f being FinSequence of () st f is_sequence_on G & i in dom f & i + 1 in dom f & n in dom G & f /. i in rng (Line (G,n)) & not f /. (i + 1) in rng (Line (G,n)) holds
for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1

let G be Go-board; :: thesis: for f being FinSequence of () st f is_sequence_on G & i in dom f & i + 1 in dom f & n in dom G & f /. i in rng (Line (G,n)) & not f /. (i + 1) in rng (Line (G,n)) holds
for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1

let f be FinSequence of (); :: thesis: ( f is_sequence_on G & i in dom f & i + 1 in dom f & n in dom G & f /. i in rng (Line (G,n)) & not f /. (i + 1) in rng (Line (G,n)) implies for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 )

assume that
A1: f is_sequence_on G and
A2: i in dom f and
A3: i + 1 in dom f and
A4: ( n in dom G & f /. i in rng (Line (G,n)) ) ; :: thesis: ( f /. (i + 1) in rng (Line (G,n)) or for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 )

consider j1, j2 being Nat such that
A5: [j1,j2] in Indices G and
A6: f /. (i + 1) = G * (j1,j2) by A1, A3;
A7: Indices G = [:(dom G),(Seg ()):] by MATRIX_0:def 4;
then A8: j1 in dom G by ;
consider i1, i2 being Nat such that
A9: [i1,i2] in Indices G and
A10: f /. i = G * (i1,i2) by A1, A2;
A11: i2 in Seg () by ;
len (Line (G,i1)) = width G by MATRIX_0:def 7;
then A12: i2 in dom (Line (G,i1)) by ;
(Line (G,i1)) . i2 = f /. i by ;
then A13: f /. i in rng (Line (G,i1)) by ;
i1 in dom G by ;
then i1 = n by A4, A13, Th2;
then A14: |.(n - j1).| + |.(i2 - j2).| = 1 by A1, A2, A3, A9, A10, A5, A6;
A15: j2 in Seg () by ;
len (Line (G,j1)) = width G by MATRIX_0:def 7;
then A16: j2 in dom (Line (G,j1)) by ;
A17: (Line (G,j1)) . j2 = f /. (i + 1) by ;
then A18: f /. (i + 1) in rng (Line (G,j1)) by ;
now :: thesis: ( f /. (i + 1) in rng (Line (G,n)) or for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 )
per cases ( ( |.(n - j1).| = 1 & i2 = j2 ) or ( |.(i2 - j2).| = 1 & n = j1 ) ) by ;
suppose ( |.(n - j1).| = 1 & i2 = j2 ) ; :: thesis: ( f /. (i + 1) in rng (Line (G,n)) or for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 )

hence ( f /. (i + 1) in rng (Line (G,n)) or for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 ) by A8, A18, Th2; :: thesis: verum
end;
suppose ( |.(i2 - j2).| = 1 & n = j1 ) ; :: thesis: ( f /. (i + 1) in rng (Line (G,n)) or for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 )

hence ( f /. (i + 1) in rng (Line (G,n)) or for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 ) by ; :: thesis: verum
end;
end;
end;
hence ( f /. (i + 1) in rng (Line (G,n)) or for k being Nat st f /. (i + 1) in rng (Line (G,k)) & k in dom G holds
|.(n - k).| = 1 ) ; :: thesis: verum