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 (0,G) iff ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) )
let l, r be Element of REAL d; :: thesis: for G being Grating of d holds
( cell (l,r) in cells (0,G) iff ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) )
let G be Grating of d; :: thesis: ( cell (l,r) in cells (0,G) iff ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) )
hereby :: thesis: ( l = r & ( for i being Element of Seg d holds l . i in G . i ) implies cell (l,r) in cells (0,G) )
assume cell (l,r) in cells (0,G) ; :: thesis: ( l = r & ( for i being Element of Seg d holds l . i in G . i ) )
then consider x being Element of REAL d such that
A1: cell (l,r) = cell (x,x) and
A2: for i being Element of Seg d holds x . i in G . i by Th34;
A3: for i being Element of Seg d holds x . i <= x . i ;
then l = x by A1, Th28;
hence ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) by A1, A2, A3, Th28; :: thesis: verum
end;
thus ( l = r & ( for i being Element of Seg d holds l . i in G . i ) implies cell (l,r) in cells (0,G) ) by Th34; :: thesis: verum