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 Th42:
  k <= d & k9 <= d & cell(l,r) in cells(k,G) & cell(l9,r9) in cells(k9,G) &
  cell(l,r) c= cell(l9,r9) implies for i holds l.i = l9.i & r.i = r9.i or
  l.i = l9.i & r.i = l9.i or l.i = r9.i & r.i = r9.i or
  l.i <= r.i & r9.i < l9.i & r9.i <= l.i & r.i <= l9.i
proof
  assume that
A1: k <= d and
A2: k9 <= d and
A3: cell(l,r) in cells(k,G) and
A4: cell(l9,r9) in cells(k9,G);
  assume
A5: cell(l,r) c= cell(l9,r9);
  let i;
  per cases by A2,A4,Th31;
  suppose
A6: for i holds l9.i < r9.i & [l9.i,r9.i] is Gap of G.i or
    l9.i = r9.i & l9.i in G.i;
    then
A7: for i holds l9.i <= r9.i;
    then
A8: l9.i <= l.i by A5,Th25;
A9: l.i <= r.i by A5,A7,Th25;
A10: r.i <= r9.i by A5,A7,Th25;
A11: l9.i <= r.i by A8,A9,XXREAL_0:2;
A12: l.i <= r9.i by A9,A10,XXREAL_0:2;
    thus thesis
    proof
      per cases by A6;
      suppose
A13:    [l9.i,r9.i] is Gap of G.i;
A14:    now
          assume that
A15:      l9.i <> l.i and
A16:      l.i <> r9.i;
A17:      l9.i < l.i by A8,A15,XXREAL_0:1;
A18:      l.i < r9.i by A12,A16,XXREAL_0:1;
          l.i in G.i by A1,A3,Th32;
          hence contradiction by A13,A17,A18,Th13;
        end;
        now
          assume that
A19:      l9.i <> r.i and
A20:      r.i <> r9.i;
A21:      l9.i < r.i by A11,A19,XXREAL_0:1;
A22:      r.i < r9.i by A10,A20,XXREAL_0:1;
          r.i in G.i by A1,A3,Th32;
          hence contradiction by A13,A21,A22,Th13;
        end;
        hence thesis by A9,A14,XXREAL_0:1;
      end;
      suppose l9.i = r9.i;
        hence thesis by A8,A10,A11,A12,XXREAL_0:1;
      end;
    end;
  end;
  suppose
A23: for i holds r9.i < l9.i & [l9.i,r9.i] is Gap of G.i;
    then
A24: r9.i < l9.i;
A25: [l9.i,r9.i] is Gap of G.i by A23;
    thus thesis
    proof
      per cases by A1,A3,Th31;
      suppose
A26:    for i holds r.i < l.i & [l.i,r.i] is Gap of G.i;
        then
A27:    r.i <= r9.i by A5,Th26;
A28:    l9.i <= l.i by A5,A26,Th26;
A29:    now
          assume l9.i <> l.i;
          then
A30:      l9.i < l.i by A28,XXREAL_0:1;
          l.i in G.i by A1,A3,Th32;
          hence contradiction by A24,A25,A30,Th13;
        end;
        now
          assume r.i <> r9.i;
          then
A31:      r.i < r9.i by A27,XXREAL_0:1;
          r.i in G.i by A1,A3,Th32;
          hence contradiction by A24,A25,A31,Th13;
        end;
        hence thesis by A29;
      end;
      suppose
A32:    for i holds l.i < r.i & [l.i,r.i] is Gap of G.i or
        l.i = r.i & l.i in G.i;
A33:    l.i in G.i by A1,A3,Th32;
        r.i in G.i by A1,A3,Th32;
        hence thesis by A24,A25,A32,A33,Th13;
      end;
    end;
  end;
end;
