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 Th22:
  (ex i st r.i < l.i) implies
  (x in cell(l,r) iff ex i st r.i < l.i & (x.i <= r.i or l.i <= x.i))
proof
  given i0 such that
A1: r.i0 < l.i0;
  x.i0 < l.i0 or r.i0 < x.i0 by A1,XXREAL_0:2;
  hence
  x in cell(l,r) implies ex i st r.i < l.i & (x.i <= r.i or l.i <= x.i)
  by Th20;
  thus thesis;
end;
