let i, n be Element of NAT ; :: thesis:
let G be Go-board; :: thesis:
let f be FinSequence of (TOP-REAL 2); :: thesis:
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 Seg (width G) & f /. i in rng (Col (G,n)) ) ; :: thesis:
consider j1, j2 being Element of NAT such that
A5: [j1,j2] in Indices G and
A6: f /. (i + 1) = G * (j1,j2) by A1, A3, Def11;
A7: Indices G = [:(dom G),(Seg (width G)):] by MATRIX_1:def 4;
then A8: j1 in dom G by A5, ZFMISC_1:87;
A9: j2 in Seg (width G) by A5, A7, ZFMISC_1:87;
len (Col (G,j2)) = len G by MATRIX_1:def 8;
then A10: j1 in dom (Col (G,j2)) by A8, FINSEQ_3:29;
consider i1, i2 being Element of NAT such that
A11: [i1,i2] in Indices G and
A12: f /. i = G * (i1,i2) by A1, A2, Def11;
A13: i1 in dom G by A11, A7, ZFMISC_1:87;
len (Col (G,i2)) = len G by MATRIX_1:def 8;
then A14: i1 in dom (Col (G,i2)) by A13, FINSEQ_3:29;
(Col (G,i2)) . i1 = f /. i by A12, A13, MATRIX_1:def 8;
then A15: f /. i in rng (Col (G,i2)) by A14, FUNCT_1:def 3;
i2 in Seg (width G) by A11, A7, ZFMISC_1:87;
then i2 = n by A4, A15, Th20;
then A16: (abs (i1 - j1)) + (abs (n - j2)) = 1 by A1, A2, A3, A11, A12, A5, A6, Def11;
A17: (Col (G,j2)) . j1 = f /. (i + 1) by A6, A8, MATRIX_1:def 8;
then A18: f /. (i + 1) in rng (Col (G,j2)) by A10, FUNCT_1:def 3;

per cases by A16, SEQM_3:42;

suppose ( abs (i1 - j1) = 1 & n = j2 ) ; :: thesis:

hence ( f /. (i + 1) in rng (Col (G,n)) or for k being Element of NAT st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds
abs (n - k) = 1 ) by A17, A10, FUNCT_1:def 3; :: thesis:

end;

suppose ( abs (n - j2) = 1 & i1 = j1 ) ; :: thesis:

hence ( f /. (i + 1) in rng (Col (G,n)) or for k being Element of NAT st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds
abs (n - k) = 1 ) by A9, A18, Th20; :: thesis:

end;

end;

end;

hence ( f /. (i + 1) in rng (Col (G,n)) or for k being Element of NAT st f /. (i + 1) in rng (Col (G,k)) & k in Seg (width G) holds
abs (n - k) = 1 ) ; :: thesis: