let d be non zero Nat; :: thesis: for l, r being Element of REAL d
for G being Grating of d st cell (l,r) is Cell of d,G holds
( cell (l,r) = infinite-cell G iff 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 st cell (l,r) is Cell of d,G holds
( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i )
let G be Grating of d; :: thesis: ( cell (l,r) is Cell of d,G implies ( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i ) )
assume A1: cell (l,r) is Cell of d,G ; :: thesis: ( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i )
then reconsider A = cell (l,r) as Cell of d,G ;

hereby :: thesis: ( ( for i being Element of Seg d holds r . i < l . i ) implies cell (l,r) = infinite-cell G )

assume cell (l,r) = infinite-cell G ; :: thesis: for i being Element of Seg d holds r . i < l . i
then consider l9, r9 being Element of REAL d such that
A2: cell (l,r) = cell (l9,r9) and
A3: for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) by Def10;
A4: l = l9 by A2, A3, Th28;
r = r9 by A2, A3, Th28;
hence for i being Element of Seg d holds r . i < l . i by A3, A4; :: thesis: verum

end;

set i0 = the Element of Seg d;
assume for i being Element of Seg d holds r . i < l . i ; :: thesis: cell (l,r) = infinite-cell G
then A5: r . the Element of Seg d < l . the Element of Seg d ;
A6: A = cell (l,r) ;
for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) by A1, A5, Th31;
hence cell (l,r) = infinite-cell G by A6, Def10; :: thesis: verum