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 Th4:
  for X,Y be RealNormSpace,
        V be Subset of [:X,Y:],
        D be Subset of X
  st D is open & V = [:D,the carrier of Y:]
  holds V is open
  proof
    let X,Y be RealNormSpace,
          V be Subset of [:X,Y:],
          D be Subset of X;
    assume
    A1: D is open & V = [:D,the carrier of Y:];

    for x be Point of X, y be Point of Y
    st [x,y] in V
    ex r1,r2 be Real
     st 0 < r1 & 0 < r2
      & [:Ball(x,r1),Ball(y,r2):] c= V
    proof
      let x be Point of X, y be Point of Y;
      assume [x,y] in V; then
      x in D & y in the carrier of Y by A1,ZFMISC_1:87; then
      consider r be Real such that
      A2: r > 0 & Ball(x,r) c= D by A1,NDIFF_8:20;

      [:Ball(x,r),Ball(y,r):] c= V
      proof
        let z be object;
        assume z in [:Ball(x,r),Ball(y,r):]; then
        consider qx,qy be object such that
        A4: qx in Ball(x,r) & qy in Ball(y,r)
           & z = [qx,qy] by ZFMISC_1:def 2;
        thus z in V by A1,A2,A4,ZFMISC_1:87;
      end;
      hence thesis by A2;
    end;
    hence thesis by Th3;
  end;
