
theorem Th26:
  for X be RealNormSpace
  st ex D be Subset of the carrier of X st D is dense countable
  holds X is separable
  proof
    let X be RealNormSpace;
    set Y = LinearTopSpaceNorm X;
    given D0 be Subset of the carrier of X such that
    A1: D0 is dense countable;
    reconsider D = D0 as Subset of Y by NORMSP_2:def 4;
    D is dense by A1,NORMSP_3:15; then
    A2: density Y c= card D by TOPGEN_1:def 12;
    card D c= omega by A1; then
    density Y c= omega by A2;
    hence thesis by TOPGEN_1:def 13;
  end;
