
theorem
  for X be RealNormSpace, A being Subset of X
  holds A is closed iff Cl A = A
  proof
    let X be RealNormSpace, A be Subset of X;
    reconsider A1 = A as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
    A1: Cl A1 = Cl A by EQCL1;
    A1 is closed iff A is closed by NORMSP_2:32;
    hence thesis by A1,PRE_TOPC:22;
  end;
