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
  ex li,ri st li in Gi & ri in Gi &
  ri < li & for xi st xi in Gi holds not (xi < ri or li < xi)
proof
  consider li being Element of REAL such that
A1: li in Gi and
A2: for xi st xi in Gi holds li >= xi by Th9;
  consider ri being Element of REAL such that
A3: ri in Gi and
A4: for xi st xi in Gi holds ri <= xi by Th10;
  take li,ri;
A5: ri <= li by A2,A3;
  now
    assume
A6: li = ri;
    consider x1,x2 be object such that
A7: x1 in Gi and
A8: x2 in Gi and
A9: x1 <> x2 by ZFMISC_1:def 10;
    reconsider x1,x2 as Element of REAL by A7,A8;
A10: ri <= x1 by A4,A7;
A11: x1 <= li by A2,A7;
A12: ri <= x2 by A4,A8;
A13: x2 <= li by A2,A8;
    x1 = li by A6,A10,A11,XXREAL_0:1;
    hence contradiction by A6,A9,A12,A13,XXREAL_0:1;
  end;
  hence thesis by A1,A2,A3,A4,A5,XXREAL_0:1;
end;
