
theorem
  for X be RealNormSpace, A,B being Subset of X
  st A c= B holds Cl A c= Cl B
  proof
    let X be RealNormSpace, A,B be Subset of X;
    assume
    A1: A c= B;
    reconsider A1 = A, B1 = B as Subset of LinearTopSpaceNorm X
    by NORMSP_2:def 4;
    A2: Cl A1 = Cl A by EQCL1;
    Cl B1 = Cl B by EQCL1;
    hence thesis by A1,A2,PRE_TOPC:19;
  end;
