let i, j, k be Element of NAT ; :: thesis:
let f be non constant standard special_circular_sequence; :: thesis:
assume that
A1: ( 1 <= i & i + 1 < len (GoB f) ) and
A2: ( 1 <= j & j <= width (GoB f) ) and
A3: ( 1 <= k & k + 1 < len f ) and
A4: LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg f,k and
A5: LSeg ((GoB f) * (i + 1),j),((GoB f) * (i + 2),j) = LSeg f,(k + 1) ; :: thesis:
A6: LSeg ((GoB f) * i,j),((GoB f) * (i + 1),j) = LSeg (f /. k),(f /. (k + 1)) by A3, A4, TOPREAL1:def 5;
then A7: ( ( (GoB f) * i,j = f /. k & (GoB f) * (i + 1),j = f /. (k + 1) ) or ( (GoB f) * i,j = f /. (k + 1) & (GoB f) * (i + 1),j = f /. k ) ) by SPPOL_1:25;
A8: k + (1 + 1) = (k + 1) + 1 ;
then A9: k + 2 <= len f by A3, NAT_1:13;
1 <= k + 1 by NAT_1:11;
then A10: LSeg ((GoB f) * (i + 1),j),((GoB f) * (i + 2),j) = LSeg (f /. (k + 1)),(f /. (k + 2)) by A5, A8, A9, TOPREAL1:def 5;
then A11: ( ( (GoB f) * (i + 1),j = f /. (k + 1) & (GoB f) * (i + 2),j = f /. (k + 2) ) or ( (GoB f) * (i + 1),j = f /. (k + 2) & (GoB f) * (i + 2),j = f /. (k + 1) ) ) by SPPOL_1:25;
i + (1 + 1) = (i + 1) + 1 ;
then A12: i + 2 <= len (GoB f) by A1, NAT_1:13;
i < i + 2 by XREAL_1:31;
then A13: ((GoB f) * i,j) `1 < ((GoB f) * (i + 2),j) `1 by A1, A2, A12, GOBOARD5:4;
hence f /. k = (GoB f) * i,j by A7, A10, SPPOL_1:25; :: thesis:
thus f /. (k + 1) = (GoB f) * (i + 1),j by A6, A11, A13, SPPOL_1:25; :: thesis:
thus f /. (k + 2) = (GoB f) * (i + 2),j by A6, A11, A13, SPPOL_1:25; :: thesis: