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 LMBALL1:
  for M be RealNormSpace, p be Element of M, r1,r2 be Real
  st r1 <= r2 holds
    cl_Ball(p,r1) c= cl_Ball(p,r2)
  & Ball(p,r1) c= cl_Ball(p,r2)
  & Ball(p,r1) c= Ball(p,r2)
  proof
    let M be RealNormSpace, p be Element of M,
        r1,r2 be Real;
    assume
    A1: r1 <= r2;
    thus cl_Ball(p,r1) c= cl_Ball(p,r2)
    proof
      let x be object;
      assume
      A2: x in cl_Ball(p,r1); then
      reconsider y=x as Point of M;
      ex q be Element of M st y=q & ||.p - q.|| <= r1 by A2; then
      ||.p - y.|| <= r2 by A1,XXREAL_0:2;
      hence x in cl_Ball(p,r2);
    end;
    thus Ball(p,r1) c= cl_Ball(p,r2)
    proof
      let x be object;
      assume
      A3: x in Ball(p,r1); then
      reconsider y = x as Point of M;
      ex q be Element of M st y = q & ||.p - q.|| < r1 by A3; then
      ||.p - y.|| <= r2 by A1,XXREAL_0:2;
      hence x in cl_Ball(p,r2);
    end;
    let x be object;
    assume
    A4: x in Ball(p,r1); then
    reconsider y = x as Point of M;
    ex q be Element of M st y = q & ||.p - q.|| < r1 by A4; then
    ||.p - y.|| < r2 by A1,XXREAL_0:2;
    hence x in Ball(p,r2);
  end;
