
theorem EQCL2:
  for X be RealNormSpace,
      Y be Subset of X,
      Z be Subset of LinearTopSpaceNorm X
  st Y = Z
  holds Y is dense iff Z is dense
  proof
    let X be RealNormSpace,
        Y be Subset of X,
        Z be Subset of LinearTopSpaceNorm X;
    assume
    A1: Y = Z;
    hereby
      assume Y is dense; then
      Cl Z = [#] X by A1,EQCL1
          .= [#] LinearTopSpaceNorm X by NORMSP_2:def 4;
      hence Z is dense;
    end;
    assume Z is dense; then
    Cl Y = [#] (LinearTopSpaceNorm X) by A1,EQCL1
        .= [#]X by NORMSP_2:def 4;
    hence Y is dense;
  end;
