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 Th26:
  (for i holds r.i < l.i) implies (cell(l,r) c= cell(l9,r9) iff
  for i holds r.i <= r9.i & r9.i < l9.i & l9.i <= l.i)
proof
  assume
A1: for i holds r.i < l.i;
  thus cell(l,r) c= cell(l9,r9) implies
  for i holds r.i <= r9.i & r9.i < l9.i & l9.i <= l.i
  proof
    assume
A2: cell(l,r) c= cell(l9,r9);
A3: for i holds r9.i < l9.i
    proof
      let i0;
      assume
A4:   l9.i0 <= r9.i0;
      defpred P[Element of Seg d,Element of REAL] means
      ($1 = i0 implies l.$1 < $2 & r9.$1 < $2) &
      (r9.$1 < l9.$1 implies 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 i = i0 & r9.i < l9.i;
          hence thesis by A4;
        end;
        suppose
A6:       i <> i0;
          r9.i < l9.i implies
           ex xi being Element of REAL st r9.i < xi & xi < l9.i by Th1;
          hence thesis by A6;
        end;
        suppose
A7:       l9.i <= r9.i;
          ex xi being Element of REAL st l.i < xi & r9.i < xi by Th2;
          hence thesis by A7;
        end;
      end;
      consider x being Function of Seg d,REAL such that
A8:   for i holds P[i,x.i] from FUNCT_2:sch 3(A5);
      reconsider x as Element of REAL d by Def3;
A9:   r.i0 < l.i0 by A1;
      x.i0 <= r.i0 or l.i0 <= x.i0 by A8;
      then
A10:  x in cell(l,r) by A9;
      per cases by A2,A10,Th20;
      suppose for i holds l9.i <= x.i & x.i <= r9.i;
        then x.i0 <= r9.i0;
        hence contradiction by A8;
      end;
      suppose ex i st r9.i < l9.i & (x.i <= r9.i or l9.i <= x.i);
        hence contradiction by A8;
      end;
    end;
    let i0;
    hereby
      assume
A11:  r9.i0 < r.i0;
      defpred P[Element of Seg d,Element of REAL] means
      r9.$1 < $2 & $2 < l9.$1 & ($1 = i0 implies $2 < r.$1);
A12:  for i ex xi being Element of REAL st P[i,xi]
      proof
        let i;
        per cases;
        suppose
A13:      i = i0 & l9.i <= r.i;
          r9.i < l9.i by A3;
          then consider xi being Element of REAL such that
A14:      r9.i < xi and
A15:      xi < l9.i by Th1;
          xi < r.i by A13,A15,XXREAL_0:2;
          hence thesis by A14,A15;
        end;
        suppose
A16:      i = i0 & r.i <= l9.i;
          then consider xi being Element of REAL such that
A17:      r9.i < xi and
A18:      xi < r.i by A11,Th1;
          xi < l9.i by A16,A18,XXREAL_0:2;
          hence thesis by A17,A18;
        end;
        suppose
A19:      i <> i0;
          r9.i < l9.i by A3;
          then ex xi being Element of REAL st ( r9.i < xi)&( xi < l9.i) by Th1;
          hence thesis by A19;
        end;
      end;
      consider x being Function of Seg d,REAL such that
A20:  for i holds P[i,x.i] from FUNCT_2:sch 3(A12);
      reconsider x as Element of REAL d by Def3;
A21:  r.i0 < l.i0 by A1;
      x.i0 <= r.i0 or l.i0 <= x.i0 by A20;
      then
A22:  x in cell(l,r) by A21;
      not (l9.i0 <= x.i0 & x.i0 <= r9.i0) by A3,XXREAL_0:2;
      then ex i st r9.i < l9.i & (x.i <= r9.i or l9.i <= x.i) by A2,A22,Th20;
      hence contradiction by A20;
    end;
    thus r9.i0 < l9.i0 by A3;
    hereby
      assume
A23:  l9.i0 > l.i0;
      defpred R[Element of Seg d,Element of REAL] means
      l9.$1 > $2 & $2 > r9.$1 & ($1 = i0 implies $2 > l.$1);
A24:  for i ex xi being Element of REAL st R[i,xi]
      proof
        let i;
        per cases;
        suppose
A25:      i = i0 & r9.i >= l.i;
          l9.i > r9.i by A3;
          then consider xi being Element of REAL such that
A26:      r9.i < xi and
A27:      xi < l9.i by Th1;
          xi > l.i by A25,A26,XXREAL_0:2;
          hence thesis by A26,A27;
        end;
        suppose
A28:      i = i0 & l.i >= r9.i;
          then consider xi being Element of REAL such that
A29:      l.i < xi and
A30:      xi < l9.i by A23,Th1;
          xi > r9.i by A28,A29,XXREAL_0:2;
          hence thesis by A29,A30;
        end;
        suppose
A31:      i <> i0;
          l9.i > r9.i by A3;
          then ex xi being Element of REAL st ( r9.i < xi)&( xi < l9.i) by Th1;
          hence thesis by A31;
        end;
      end;
      consider x being Function of Seg d,REAL such that
A32:  for i holds R[i,x.i] from FUNCT_2:sch 3(A24);
      reconsider x as Element of REAL d by Def3;
A33:  l.i0 > r.i0 by A1;
      x.i0 >= l.i0 or r.i0 >= x.i0 by A32;
      then
A34:  x in cell(l,r) by A33;
      not (r9.i0 >= x.i0 & x.i0 >= l9.i0) by A3,XXREAL_0:2;
      then ex i st l9.i > r9.i & (x.i <= r9.i or l9.i <= x.i) by A2,A34,Th20;
      hence contradiction by A32;
    end;
  end;
  assume
A35: for i holds r.i <= r9.i & r9.i < l9.i & l9.i <= l.i;
  let x be object;
  assume
A36: x in cell(l,r);
  then reconsider x as Element of REAL d;
  set i0 = the Element of Seg d;
A37: r.i0 <= r9.i0 by A35;
  r9.i0 < l9.i0 by A35;
  then
A38: r.i0 < l9.i0 by A37,XXREAL_0:2;
  l9.i0 <= l.i0 by A35;
  then r.i0 < l.i0 by A38,XXREAL_0:2;
  then x.i0 < l.i0 or r.i0 < x.i0 by XXREAL_0:2;
  then consider i such that
  r.i < l.i and
A39: x.i <= r.i or l.i <= x.i by A36,Th20;
A40: r.i <= r9.i by A35;
A41: l9.i <= l.i by A35;
A42: r9.i < l9.i by A35;
  x.i <= r9.i or l9.i <= x.i by A39,A40,A41,XXREAL_0:2;
  hence thesis by A42;
end;
