
theorem
  for X be RealNormSpace, A be Subset of X
  holds
    A is dense
  iff
    for x be Point of X ex seq be sequence of X
    st rng seq c= A & seq is convergent & lim seq = x
  proof
    let X be RealNormSpace, A be Subset of X;
    thus A is dense implies for x be Point of X ex seq be sequence of X
      st rng seq c= A & seq is convergent & lim seq = x by EQCL3;
    assume
    A1: for x be Point of X ex seq be sequence of X
    st rng seq c= A & seq is convergent & lim seq = x;
    for z be object st z in [#] X holds z in Cl A
    proof
      let z be object;
      assume z in [#] X; then
      reconsider x = z as Point of X;
      ex seq be sequence of X
      st rng seq c= A & seq is convergent & lim seq = x by A1;
      hence thesis by EQCL3;
    end; then
    [#] X c= Cl A; then
    [#] X = Cl A;
    hence thesis;
  end;
