
theorem
  for X be RealNormSpace, R,S being Subset of X
  st R is dense & R c= S
  holds S is dense
  proof
    let X be RealNormSpace, R,S be Subset of X;
    reconsider R1 = R, S1 = S as Subset of LinearTopSpaceNorm X
    by NORMSP_2:def 4;
    assume R is dense & R c= S; then
    R1 is dense & R1 c= S1 by EQCL2; then
    S1 is dense by TOPS_1:44;
    hence S is dense by EQCL2;
end;
