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 Th3:
  for X,Y be RealNormSpace,
        V be Subset of [:X,Y:]
  holds
    V is open
  iff
   (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,Y be RealNormSpace,
          V be Subset of [:X,Y:];
    hereby
      assume
      A1: V is open;
      let x be Point of X, y be Point of Y;
      assume [x,y] in V; then
      consider r be Real such that
      A2: 0 < r & [:Ball(x,r),Ball(y,r):] c= V by A1,NDIFF_8:23;
      thus
      ex r1,r2 be Real
       st 0 < r1 & 0 < r2
        & [:Ball(x,r1),Ball(y,r2):] c= V by A2;
    end;

    assume
    A3: 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;

    for z be Point of [:X,Y:] st z in V
    ex s be Real st s > 0 & Ball(z,s) c= V
    proof
      let z be Point of [:X,Y:];
      assume
      A4: z in V;
      consider x be Point of X, y be Point of Y such that
      A5: z = [x,y] by PRVECT_3:18;
      consider r1,r2 be Real such that
      A6: 0 < r1 & 0 < r2
        & [:Ball(x,r1),Ball(y,r2):] c= V by A3,A4,A5;

      consider s be Real such that
      A7: s = min(r1, r2) & s > 0 and
      A8: Ball(z,s) c= [:Ball(x,r1), Ball(y,r2):] by A5,A6,Th2;

      take s;
      thus thesis by A6,A7,A8,XBOOLE_1:1;
    end;
    hence V is open by NDIFF_8:20;
  end;
