let f be non constant standard special_circular_sequence; :: thesis: for k being Nat st 1 <= k & k + 1 <= len f holds
ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )

let k be Nat; :: thesis: ( 1 <= k & k + 1 <= len f implies ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) ) )

assume that
A1: 1 <= k and
A2: k + 1 <= len f ; :: thesis: ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )

A3: f is_sequence_on GoB f by GOBOARD5:def 5;
k <= len f by ;
then A4: k in dom f by ;
then consider i1, j1 being Nat such that
A5: [i1,j1] in Indices (GoB f) and
A6: f /. k = (GoB f) * (i1,j1) by A3;
1 <= k + 1 by NAT_1:11;
then A7: k + 1 in dom f by ;
then consider i2, j2 being Nat such that
A8: [i2,j2] in Indices (GoB f) and
A9: f /. (k + 1) = (GoB f) * (i2,j2) by A3;
1 <= i1 by ;
then A10: (i1 -' 1) + 1 = i1 by XREAL_1:235;
1 <= j1 by ;
then A11: (j1 -' 1) + 1 = j1 by XREAL_1:235;
reconsider i19 = i1, i29 = i2, j19 = j1, j29 = j2 as Element of REAL by XREAL_0:def 1;
|.(i1 - i2).| + |.(j1 - j2).| = 1 by A3, A4, A5, A6, A7, A8, A9;
then A12: ( ( |.(i19 - i29).| = 1 & j1 = j2 ) or ( |.(j19 - j29).| = 1 & i1 = i2 ) ) by SEQM_3:42;
A13: i1 <= len (GoB f) by ;
A14: j1 <= width (GoB f) by ;
per cases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by ;
suppose A15: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )

take i1 -' 1 ; :: thesis: ex j being Nat st
( i1 -' 1 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),(i1 -' 1),j) = left_cell (f,k) )

take j1 ; :: thesis: ( i1 -' 1 <= len (GoB f) & j1 <= width (GoB f) & cell ((GoB f),(i1 -' 1),j1) = left_cell (f,k) )
i1 -' 1 <= i1 by NAT_D:35;
hence i1 -' 1 <= len (GoB f) by ; :: thesis: ( j1 <= width (GoB f) & cell ((GoB f),(i1 -' 1),j1) = left_cell (f,k) )
thus j1 <= width (GoB f) by ; :: thesis: cell ((GoB f),(i1 -' 1),j1) = left_cell (f,k)
thus cell ((GoB f),(i1 -' 1),j1) = left_cell (f,k) by A1, A2, A5, A6, A8, A9, A10, A15, GOBOARD5:27; :: thesis: verum
end;
suppose A16: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )

take i1 ; :: thesis: ex j being Nat st
( i1 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i1,j) = left_cell (f,k) )

take j1 ; :: thesis: ( i1 <= len (GoB f) & j1 <= width (GoB f) & cell ((GoB f),i1,j1) = left_cell (f,k) )
thus i1 <= len (GoB f) by ; :: thesis: ( j1 <= width (GoB f) & cell ((GoB f),i1,j1) = left_cell (f,k) )
thus j1 <= width (GoB f) by ; :: thesis: cell ((GoB f),i1,j1) = left_cell (f,k)
thus cell ((GoB f),i1,j1) = left_cell (f,k) by A1, A2, A5, A6, A8, A9, A11, A16, GOBOARD5:28; :: thesis: verum
end;
suppose A17: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )

take i2 ; :: thesis: ex j being Nat st
( i2 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i2,j) = left_cell (f,k) )

take j1 -' 1 ; :: thesis: ( i2 <= len (GoB f) & j1 -' 1 <= width (GoB f) & cell ((GoB f),i2,(j1 -' 1)) = left_cell (f,k) )
thus i2 <= len (GoB f) by ; :: thesis: ( j1 -' 1 <= width (GoB f) & cell ((GoB f),i2,(j1 -' 1)) = left_cell (f,k) )
j1 -' 1 <= j1 by NAT_D:35;
hence j1 -' 1 <= width (GoB f) by ; :: thesis: cell ((GoB f),i2,(j1 -' 1)) = left_cell (f,k)
thus cell ((GoB f),i2,(j1 -' 1)) = left_cell (f,k) by A1, A2, A5, A6, A8, A9, A11, A17, GOBOARD5:29; :: thesis: verum
end;
suppose A18: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )

take i1 ; :: thesis: ex j being Nat st
( i1 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i1,j) = left_cell (f,k) )

take j2 ; :: thesis: ( i1 <= len (GoB f) & j2 <= width (GoB f) & cell ((GoB f),i1,j2) = left_cell (f,k) )
thus i1 <= len (GoB f) by ; :: thesis: ( j2 <= width (GoB f) & cell ((GoB f),i1,j2) = left_cell (f,k) )
thus j2 <= width (GoB f) by ; :: thesis: cell ((GoB f),i1,j2) = left_cell (f,k)
thus cell ((GoB f),i1,j2) = left_cell (f,k) by A1, A2, A5, A6, A8, A9, A10, A18, GOBOARD5:30; :: thesis: verum
end;
end;