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

hereby :: thesis: ( ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . 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 & [(l . i),(r . i)] is Gap of G . i )
then consider l', r' being Element of REAL d such that
A1: cell l,r = cell l',r' and
A2: for i being Element of Seg d holds
( r' . i < l' . i & [(l' . i),(r' . i)] is Gap of G . i ) by Def11;
A3: l = l' by A1, A2, Th32;
r = r' by A1, A2, Th32;
hence for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) by A2, A3; :: thesis: verum

end;

assume A4: for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ; :: thesis: cell l,r = infinite-cell G
then cell l,r is Cell of d,G by Th34;
hence cell l,r = infinite-cell G by A4, Def11; :: thesis: verum