reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;
reserve d for non zero Nat;
reserve i,i0,i1 for Element of Seg d;
reserve l,r,l9,r9,l99,r99,x,x9,l1,r1,l2,r2 for Element of REAL d;
reserve Gi for non trivial finite Subset of REAL;
reserve li,ri,li9,ri9,xi,xi9 for Real;
reserve G for Grating of d;

theorem Th8:
  x in product G iff for i holds x.i in G.i
proof
  x is Function of Seg d,REAL by Def3;
  then
A1: dom x = Seg d by FUNCT_2:def 1;
A2: dom G = Seg d by FUNCT_2:def 1;
  hence x in product G implies for i holds x.i in G.i by CARD_3:9;
  assume for i holds x.i in G.i;
  then for i being object st i in Seg d holds x.i in G.i;
  hence thesis by A1,A2,CARD_3:9;
