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 Th24:
  cell(x,x) = {x}
proof
  for x9 being object holds x9 in cell(x,x) iff x9 = x
  proof
    let x9 be object;
    thus x9 in cell(x,x) implies x9 = x
    proof
      assume
A1:   x9 in cell(x,x);
      then reconsider x,x9 as Function of Seg d,REAL by Def3;
      now
        let i;
A2:     for i holds x.i <= x.i;
        then
A3:     x.i <= x9.i by A1,Th21;
        x9.i <= x.i by A1,A2,Th21;
        hence x9.i = x.i by A3,XXREAL_0:1;
      end;
      hence thesis by FUNCT_2:63;
    end;
    thus thesis by Th23;
  end;
  hence thesis by TARSKI:def 1;
end;
