
theorem LMQ23:
  for V be RealNormSpace, W be Subspace of V, v be VECTOR of V holds
  0 <= lower_bound NormVSets(V,W,v) & lower_bound NormVSets(V,W,v) <= ||.v.||
  proof
    let V be RealNormSpace, W be Subspace of V, v be VECTOR of V;
    for r being ExtReal st r in NormVSets(V,W,v) holds 0 <= r
    proof
      let r be ExtReal;
      assume r in NormVSets(V,W,v); then
      consider x be VECTOR of V such that
      A1: r = ||.x.|| & x in v + W;
      thus thesis by A1;
    end; then
    0 is LowerBound of NormVSets(V,W,v) by XXREAL_2:def 2;
    hence 0 <= lower_bound NormVSets(V,W,v) by XXREAL_2:def 4;
    v in v + W by RLSUB_1:43; then
    A2: ||.v.|| in NormVSets(V,W,v);
    inf NormVSets(V,W,v) is LowerBound of NormVSets(V,W,v) by XXREAL_2:def 4;
    hence thesis by A2,XXREAL_2:def 2;
  end;
