let k be Element of NAT ; :: thesis: for f being standard special_circular_sequence st 1 <= k & k + 1 <= len f holds
left_cell f,k,(GoB f) = left_cell f,k
let f be standard special_circular_sequence; :: thesis: ( 1 <= k & k + 1 <= len f implies left_cell f,k,(GoB f) = left_cell f,k )
assume that
A1: 1 <= k and
A2: k + 1 <= len f ; :: thesis: left_cell f,k,(GoB f) = left_cell f,k
set G = GoB f;
A3: f is_sequence_on GoB f by GOBOARD5:def 5;
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (GoB f) & [i2,j2] in Indices (GoB f) & f /. k = (GoB f) * i1,j1 & f /. (k + 1) = (GoB f) * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & left_cell f,k = cell (GoB f),(i1 -' 1),j1 ) & not ( i1 + 1 = i2 & j1 = j2 & left_cell f,k = cell (GoB f),i1,j1 ) & not ( i1 = i2 + 1 & j1 = j2 & left_cell f,k = cell (GoB f),i2,(j2 -' 1) ) holds
( i1 = i2 & j1 = j2 + 1 & left_cell f,k = cell (GoB f),i1,j2 )

proof

1 <= k + 1 by NAT_1:11;
then A4: k + 1 in dom f by A2, FINSEQ_3:27;
let i1, j1, i2, j2 be Element of NAT ; :: thesis: ( [i1,j1] in Indices (GoB f) & [i2,j2] in Indices (GoB f) & f /. k = (GoB f) * i1,j1 & f /. (k + 1) = (GoB f) * i2,j2 & not ( i1 = i2 & j1 + 1 = j2 & left_cell f,k = cell (GoB f),(i1 -' 1),j1 ) & not ( i1 + 1 = i2 & j1 = j2 & left_cell f,k = cell (GoB f),i1,j1 ) & not ( i1 = i2 + 1 & j1 = j2 & left_cell f,k = cell (GoB f),i2,(j2 -' 1) ) implies ( i1 = i2 & j1 = j2 + 1 & left_cell f,k = cell (GoB f),i1,j2 ) )
assume A5: ( [i1,j1] in Indices (GoB f) & [i2,j2] in Indices (GoB f) & f /. k = (GoB f) * i1,j1 & f /. (k + 1) = (GoB f) * i2,j2 ) ; :: thesis: ( ( i1 = i2 & j1 + 1 = j2 & left_cell f,k = cell (GoB f),(i1 -' 1),j1 ) or ( i1 + 1 = i2 & j1 = j2 & left_cell f,k = cell (GoB f),i1,j1 ) or ( i1 = i2 + 1 & j1 = j2 & left_cell f,k = cell (GoB f),i2,(j2 -' 1) ) or ( i1 = i2 & j1 = j2 + 1 & left_cell f,k = cell (GoB f),i1,j2 ) )
k < len f by A2, NAT_1:13;
then k in dom f by A1, FINSEQ_3:27;
then (abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A3, A5, A4, GOBOARD1:def 11;
then A6: ( ( abs (i1 - i2) = 1 & j1 = j2 ) or ( abs (j1 - j2) = 1 & i1 = i2 ) ) by GOBOARD1:2;
left_cell f,k = left_cell f,k ;
then A7: ( ( i1 = i2 & j1 + 1 = j2 & left_cell f,k = cell (GoB f),(i1 -' 1),j1 ) or ( i1 + 1 = i2 & j1 = j2 & left_cell f,k = cell (GoB f),i1,j1 ) or ( i1 = i2 + 1 & j1 = j2 & left_cell f,k = cell (GoB f),i2,(j2 -' 1) ) or ( i1 = i2 & j1 = j2 + 1 & left_cell f,k = cell (GoB f),i1,j2 ) ) by A1, A2, A5, GOBOARD5:def 7;

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 A6, GOBOARD1:1;

case ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: left_cell f,k = cell (GoB f),(i1 -' 1),j1

hence left_cell f,k = cell (GoB f),(i1 -' 1),j1 by A7; :: thesis: verum

end;

case ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: left_cell f,k = cell (GoB f),i1,j1

hence left_cell f,k = cell (GoB f),i1,j1 by A7; :: thesis: verum

end;

case ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: left_cell f,k = cell (GoB f),i2,(j2 -' 1)

hence left_cell f,k = cell (GoB f),i2,(j2 -' 1) by A7; :: thesis: verum

end;

case ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: left_cell f,k = cell (GoB f),i1,j2

hence left_cell f,k = cell (GoB f),i1,j2 by A7; :: thesis: verum

end;

end;

end;

hence left_cell f,k,(GoB f) = left_cell f,k by A1, A2, A3, GOBRD13:def 3; :: thesis: verum