
theorem
  for X be RealNormSpace holds
  X is separable iff ex seq be sequence of X st rng seq is dense
  proof
    let X be RealNormSpace;
    set Y = LinearTopSpaceNorm X;
    consider B being Subset of Y such that
    A1: B is dense & density Y = card B by TOPGEN_1:def 12;
    hereby
      assume
      A2: X is separable;
      reconsider A = B as Subset of X by NORMSP_2:def 4;
      A is dense by A1,EQCL2; then
      A3: A is non empty by NONEMP;
      A is countable by A1,A2,CARD_3:def 14,TOPGEN_1:def 13; then
      consider f be Function of omega, A such that
      A4: rng f = A by A3,CARD_3:96;
      reconsider seq = f as Function of NAT, the carrier of X
      by A3,A4,FUNCT_2:6;
      reconsider seq as sequence of X;
      rng seq is dense by A1,A4,EQCL2;
      hence ex seq be sequence of X st rng seq is dense;
    end;
    given seq be sequence of X such that
    A5: rng seq is dense;
    reconsider D = rng seq as Subset of Y by NORMSP_2:def 4;
    D is dense by A5,EQCL2; then
    A6: density Y c= card D by TOPGEN_1:def 12;
    dom seq = NAT by FUNCT_2:def 1; then
    card D c= omega by CARD_3:93,def 14; then
    density Y c= omega by A6;
    hence thesis by TOPGEN_1:def 13;
  end;
