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 LMBALL:
  for p be Element of S, r be Real holds
    Ball(p,r) = {y where y is Point of S: ||.y - p.|| < r}
  proof
    let p be Element of S,r be Real;
    deffunc F(object) = $1;
    defpred P1[Element of S] means ||.p - $1.|| < r;
    defpred P2[Element of S] means ||.$1 - p.|| < r;
    A1: for v being Element of the carrier of S holds
        (P1[v] iff P2[v]) by NORMSP_1:7;
    {F(y) where y is Element of the carrier of S: P1[y]}
    = {F(y) where y is Element of the carrier of S: P2[y]}
      from FRAENKEL:sch 3(A1);
    hence thesis;
  end;
