let d be non zero Nat; :: thesis: for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) in cells (d,G) iff ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )

let l, r be Element of REAL d; :: thesis: for G being Grating of d holds
( cell (l,r) in cells (d,G) iff ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )

let G be Grating of d; :: thesis: ( cell (l,r) in cells (d,G) iff ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
hereby :: thesis: ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) implies cell (l,r) in cells (d,G) )
assume cell (l,r) in cells (d,G) ; :: thesis: ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) )
then consider l9, r9 being Element of REAL d such that
A1: cell (l,r) = cell (l9,r9) and
A2: for i being Element of Seg d holds [(l9 . i),(r9 . i)] is Gap of G . i and
A3: ( for i being Element of Seg d holds l9 . i < r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) by Th36;
A4: ( for i being Element of Seg d holds l9 . i <= r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) by A3;
then A5: l = l9 by ;
r = r9 by A1, A4, Th28;
hence ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) by A2, A3, A5; :: thesis: verum
end;
thus ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) implies cell (l,r) in cells (d,G) ) by Th36; :: thesis: verum