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 Th28:
  (for i holds l.i <= r.i) or (for i holds l.i > r.i) implies
  (cell(l,r) = cell(l9,r9) iff l = l9 & r = r9)
proof
  assume
A1: (for i holds l.i <= r.i) or for i holds l.i > r.i;
  thus cell(l,r) = cell(l9,r9) implies l = l9 & r = r9
  proof
    assume
A2: cell(l,r) = cell(l9,r9);
    per cases by A1;
    suppose
A3:   for i holds l.i <= r.i;
      then
A4:   for i holds l.i <= l9.i & l9.i <= r9.i & r9.i <= r.i by A2,Th25;
      reconsider l,r,l9,r9 as Function of Seg d,REAL by Def3;
A5:   now
        let i;
A6:     l.i <= l9.i by A2,A3,Th25;
        l9.i <= l.i by A2,A4,Th25;
        hence l.i = l9.i by A6,XXREAL_0:1;
      end;
      now
        let i;
A7:     r.i <= r9.i by A2,A4,Th25;
        r9.i <= r.i by A2,A3,Th25;
        hence r.i = r9.i by A7,XXREAL_0:1;
      end;
      hence thesis by A5,FUNCT_2:63;
    end;
    suppose
A8:   for i holds l.i > r.i;
      then
A9:   for i holds r.i <= r9.i & r9.i < l9.i & l9.i <= l.i by A2,Th26;
      reconsider l,r,l9,r9 as Function of Seg d,REAL by Def3;
A10:  now
        let i;
A11:    l.i <= l9.i by A2,A9,Th26;
        l9.i <= l.i by A2,A8,Th26;
        hence l.i = l9.i by A11,XXREAL_0:1;
      end;
      now
        let i;
A12:    r.i <= r9.i by A2,A8,Th26;
        r9.i <= r.i by A2,A9,Th26;
        hence r.i = r9.i by A12,XXREAL_0:1;
      end;
      hence thesis by A10,FUNCT_2:63;
    end;
  end;
  thus thesis;
end;
