
theorem
  for X be RealNormSpace, A being Subset of X
  holds A is open iff Cl([#](X) \ A) = [#](X) \ 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: [#](X) = [#](LinearTopSpaceNorm X) by NORMSP_2:def 4;
    A2: Cl([#](LinearTopSpaceNorm X) \ A1) = Cl([#]X \ A) by A1,EQCL1;
    A1 is open iff A is open by NORMSP_2:33;
    hence thesis by A1,A2,PRE_TOPC:23;
  end;
