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
  for X,Y be RealNormSpace,
        x be Point of X,
        y be Point of Y,
        V be Subset of [:X,Y:]
  st V is open & [x,y] in V
  ex r be Real st 0 < r & [:Ball(x,r),Ball(y,r):] c= V
  proof
    let X,Y be RealNormSpace,
          x be Point of X,
          y be Point of Y,
          V be Subset of [:X,Y:];
    assume
A1: V is open & [x,y] in V;
    reconsider z = [x,y] as Point of [:X,Y:];
    consider r be Real such that
A2: r>0 & Ball(z,r) c= V by A1,NORMSP27;
    consider r2 be Real such that
A3: 0 < r2 & r2 < r
      & [:Ball(x,r2),Ball(y,r2):] c= Ball(z,r) by A2,NORMSP31;
    take r2;
    thus 0 < r2 by A3;
    thus [:Ball(x,r2),Ball(y,r2):] c= V by A2,A3,XBOOLE_1:1;
  end;
