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 LMBALL1X:
  for M be RealNormSpace, p be Element of M, r1,r2 be Real
    st r1 < r2 holds
  cl_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;
    assume not cl_Ball(p,r1) c= Ball(p,r2); then
    consider x be object such that
    A2: x in cl_Ball(p,r1) & not x in Ball(p,r2) by TARSKI:def 3;
    reconsider x as Point of M by A2;
    A3: ex q be Element of M st x = q & ||.p - q.|| <= r1 by A2;
    r2 <= ||.p - x.|| by A2;
    hence contradiction by A1,A3,XXREAL_0:2;
  end;
