let k be Element of NAT ; :: thesis:
let D be set ; :: thesis:
let f be FinSequence of D; :: thesis:
let G be Matrix of D; :: thesis:
assume that
A1: 1 <= k and
A2: k + 1 <= len f and
A3: f is_sequence_on G ; :: thesis:
k <= k + 1 by NAT_1:11;
then k <= len f by A2, XXREAL_0:2;
then A4: k in dom f by A1, FINSEQ_3:25;
then consider i1, j1 being Element of NAT such that
A5: [i1,j1] in Indices G and
A6: f /. k = G * (i1,j1) by A3, GOBOARD1:def 9;
k + 1 >= 1 by NAT_1:11;
then A7: k + 1 in dom f by A2, FINSEQ_3:25;
then consider i2, j2 being Element of NAT such that
A8: [i2,j2] in Indices G and
A9: f /. (k + 1) = G * (i2,j2) by A3, GOBOARD1:def 9;
A10: (abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A3, A4, A5, A6, A7, A8, A9, GOBOARD1:def 9;

( i1 - i2 = 1 or - (i1 - i2) = 1 ) by A11, ABSVALUE:def 1;
hence ( i1 = i2 + 1 or i1 + 1 = i2 ) ; :: thesis:
thus j1 = j2 by A12; :: thesis:

end;

( j1 - j2 = 1 or - (j1 - j2) = 1 ) by A14, ABSVALUE:def 1;
hence ( j1 = j2 + 1 or j1 + 1 = j2 ) ; :: thesis:
thus i1 = i2 by A13; :: thesis:

end;

hence ex i1, j1, i2, j2 being Element of NAT st
( [i1,j1] in Indices G & f /. k = G * (i1,j1) & [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) & ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) ) by A5, A6, A8, A9; :: thesis: