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 Th21:
  (for i holds l.i <= r.i) implies
  (x in cell(l,r) iff for i holds l.i <= x.i & x.i <= r.i)
proof
  assume
A1: for i holds l.i <= r.i;
  hereby
    assume x in cell(l,r);
    then (for i holds l.i <= x.i & x.i <= r.i) or
    ex i st r.i < l.i & (x.i <= r.i or l.i <= x.i) by Th20;
    hence for i holds l.i <= x.i & x.i <= r.i by A1;
  end;
  thus thesis;
end;
