let k be Nat; :: thesis: for d being non zero Nat
for l, r being Element of REAL d
for G being Grating of d st k <= d holds
( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
let d be non zero Nat; :: thesis: for l, r being Element of REAL d
for G being Grating of d st k <= d holds
( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( 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 st k <= d holds
( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( 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: ( k <= d implies ( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
assume A1: k <= d ; :: thesis: ( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )

hereby :: thesis: ( ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( 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) in cells (k,G) )

assume cell (l,r) in cells (k,G) ; :: thesis: ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
then consider l9, r9 being Element of REAL d such that
A2: cell (l,r) = cell (l9,r9) and
A3: ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) ) ) by A1, Th29;
( l = l9 & r = r9 )

proof

per cases ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) or for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) by A3;

suppose ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) ; :: thesis: ( l = l9 & r = r9 )

then for i being Element of Seg d holds l9 . i <= r9 . i ;
hence ( l = l9 & r = r9 ) by A2, Th28; :: thesis: verum

end;

suppose for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ; :: thesis: ( l = l9 & r = r9 )

hence ( l = l9 & r = r9 ) by A2, Th28; :: thesis: verum

end;

end;

end;

hence ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A3; :: thesis: verum

end;

thus ( ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( 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) in cells (k,G) ) by A1, Th29; :: thesis: verum