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 Th43:
  k < k9 & k9 <= d & cell(l,r) in cells(k,G) & cell(l9,r9) in cells(k9,G) &
  cell(l,r) c= cell(l9,r9) implies
  ex i st l.i = l9.i & r.i = l9.i or l.i = r9.i & r.i = r9.i
proof
  assume that
A1: k < k9 and
A2: k9 <= d and
A3: cell(l,r) in cells(k,G) and
A4: cell(l9,r9) in cells(k9,G);
A5: k + 0 < d by A1,A2,XXREAL_0:2;
  assume
A6: cell(l,r) c= cell(l9,r9);
  consider X being Subset of Seg d such that
A7: card X = k and
A8: for i holds i in X & l.i < r.i & [l.i,r.i] is Gap of G.i or not i
  in X & l.i = r.i & l.i in G.i
  by A3,A5,Th30;
A9: d - k > 0 by A5,XREAL_1:20;
  card(Seg d \ X) = card Seg d - card X by CARD_2:44
    .= d - k by A7,FINSEQ_1:57;
  then consider i0 being object such that
A10: i0 in Seg d \ X by A9,CARD_1:27,XBOOLE_0:def 1;
  reconsider i0 as Element of Seg d by A10,XBOOLE_0:def 5;
  not i0 in X by A10,XBOOLE_0:def 5;
  then
A11: l.i0 = r.i0 by A8;
  per cases by A2,A3,A4,A5,A6,Th42;
  suppose l.i0 = l9.i0 & r.i0 = r9.i0;
    hence thesis by A11;
  end;
  suppose l.i0 = l9.i0 & r.i0 = l9.i0;
    hence thesis;
  end;
  suppose l.i0 = r9.i0 & r.i0 = r9.i0;
    hence thesis;
  end;
  suppose
A12: r9.i0 < l9.i0;
    assume
A13: for i holds (l.i <> l9.i or r.i <> l9.i) & (l.i <> r9.i or r.i <> r9.i);
    defpred P[Element of Seg d,Element of REAL] means
    l.$1 <= $2 & $2 <= r.$1 & r9.$1 < $2 & $2 < l9.$1;
A14: for i ex xi being Element of REAL st P[i,xi]
    proof
      let i;
A15:  l.i in G.i by A3,A5,Th32;
A16:  r.i in G.i by A3,A5,Th32;
A17:  r9.i < l9.i by A2,A4,A12,Th31;
A18:  [l9.i,r9.i] is Gap of G.i by A2,A4,A12,Th31;
      per cases;
      suppose
A19:    r9.i < l.i & l.i < l9.i;
         reconsider li = l.i as Element of REAL by XREAL_0:def 1;
        take li;
        thus thesis by A8,A19;
      end;
      suppose
A20:    l.i <= r9.i;
A21:    l.i >= r9.i by A15,A17,A18,Th13;
        then
A22:    l.i = r9.i by A20,XXREAL_0:1;
        then r.i <> r9.i by A13;
        then l.i < r.i by A8,A22;
        then consider xi being Element of REAL such that
A23:    l.i < xi and
A24:    xi < r.i by Th1;
        take xi;
        r.i <= l9.i by A16,A17,A18,Th13;
        hence thesis by A21,A23,A24,XXREAL_0:2;
      end;
      suppose
A25:    l9.i <= l.i;
        l9.i >= l.i by A15,A17,A18,Th13;
        then
A26:    l9.i = l.i by A25,XXREAL_0:1;
        l9.i >= r.i by A16,A17,A18,Th13;
        then l9.i = r.i by A8,A26;
        hence thesis by A13,A26;
      end;
    end;
    consider x being Function of Seg d,REAL such that
A27: for i holds P[i,x.i] from FUNCT_2:sch 3(A14);
    reconsider x as Element of REAL d by Def3;
A28: x in cell(l,r) by A27;
    for i st r9.i < l9.i holds r9.i < x.i & x.i < l9.i by A27;
    hence contradiction by A6,A12,A28,Th22;
  end;
end;
