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 Th27:
  (for i holds l.i <= r.i) & (for i holds r9.i < l9.i) implies
  (cell(l,r) c= cell(l9,r9) iff ex i st r.i <= r9.i or l9.i <= l.i)
proof
  assume
A1: for i holds l.i <= r.i;
  assume
A2: for i holds r9.i < l9.i;
  thus
  cell(l,r) c= cell(l9,r9) implies ex i st r.i <= r9.i or l9.i <= l.i
  proof
    assume
A3: cell(l,r) c= cell(l9,r9);
    assume
A4: for i holds r9.i < r.i & l.i < l9.i;
    defpred P[Element of Seg d,Element of REAL] means
    l.$1 <= $2 & $2 <= r.$1 & r9.$1 < $2 & $2 < l9.$1;
A5: for i ex xi being Element of REAL st P[i,xi]
    proof
      let i;
      per cases;
      suppose
A6:     l.i <= r9.i & l9.i <= r.i;
        r9.i < l9.i by A2;
        then consider xi being Element of REAL such that
A7:     r9.i < xi and
A8:     xi < l9.i by Th1;
        take xi;
        thus thesis by A6,A7,A8,XXREAL_0:2;
      end;
      suppose
A9:     r9.i < l.i & l9.i <= r.i;
         reconsider li = l.i as Element of REAL by XREAL_0:def 1;
        take li;
        thus thesis by A1,A4,A9;
      end;
      suppose
A10:    r.i < l9.i;
         reconsider ri = r.i as Element of REAL by XREAL_0:def 1;
        take ri;
        thus thesis by A1,A4,A10;
      end;
    end;
    consider x being Function of Seg d,REAL such that
A11: for i holds P[i,x.i] from FUNCT_2:sch 3(A5);
    reconsider x as Element of REAL d by Def3;
A12: x in cell(l,r) by A11;
    set i0 = the Element of Seg d;
    r9.i0 < l9.i0 by A2;
    then ex i st r9.i < l9.i & (x.i <= r9.i or l9.i <= x.i) by A3,A12,Th22;
    hence contradiction by A11;
  end;
  given i0 such that
A13: r.i0 <= r9.i0 or l9.i0 <= l.i0;
  let x be object;
  assume
A14: x in cell(l,r);
  then reconsider x as Element of REAL d;
A15: l.i0 <= x.i0 by A1,A14,Th21;
A16: x.i0 <= r.i0 by A1,A14,Th21;
  ex i st r9.i < l9.i & (x.i <= r9.i or l9.i <= x.i)
  proof
    take i0;
    thus thesis by A2,A13,A15,A16,XXREAL_0:2;
  end;
  hence thesis;
end;
