reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem NORMSP27:
  for X be RealNormSpace, V be Subset of X holds V is open
  iff for x be Point of X st x in V ex r be Real st r>0 & Ball(x,r) c= V
  proof
    let X be RealNormSpace, V be Subset of X;
    reconsider V0 = V as Subset of TopSpaceNorm X;
    hereby
      assume V is open; then
      A1: V0 is open by NORMSP_2:16;
      thus for x be Point of X st x in V
      ex r be Real st r>0 & Ball(x,r) c= V
      proof
        let x be Point of X;
        assume x in V; then
        consider r be Real such that
        A2: r>0
          & {y where y is Point of X: ||.x-y.|| < r} c= V0 by A1,NORMSP_2:7;
        take r;
        thus thesis by A2;
      end;
    end;
    assume
    A3: for x be Point of X st x in V
        ex r be Real st r>0 & Ball(x,r) c= V;
    for x be Point of X st x in V0
    ex r be Real st r>0
    & {y where y is Point of X: ||.x-y.|| < r} c= V0
    proof
      let x be Point of X;
      assume x in V0; then
      consider r be Real such that
      A4: r>0 & Ball(x,r) c= V by A3;
      take r;
      thus thesis by A4;
    end; then
    V0 is open by NORMSP_2:7;
    hence thesis by NORMSP_2:16;
  end;
