
theorem
  for X be RealNormSpace, R,S be Subset of X
  st R is dense & S is open
  holds Cl S = Cl(S /\ R)
  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;
    A1: Cl S = Cl S1 by EQCL1;
    A2: Cl(S1 /\ R1) = Cl(S /\ R) by EQCL1;
    assume R is dense & S is open; then
    R1 is dense & S1 is open by EQCL2,NORMSP_2:33;
    hence Cl S = Cl(S /\ R) by A1,A2,TOPS_1:46;
  end;
