
theorem Th23:
  for X be RealNormSpace,
      x be Point of DualSp X,
      M be non empty Subspace of X holds
  {||.x-m.|| where m is Point of DualSp X : m in Ort_Comp M}
    is non empty bounded_below real-membered set
proof
  let X be RealNormSpace,
      x be Point of DualSp X,
      M be non empty Subspace of X;
  set B = {||.x-m.|| where m is Point of DualSp X : m in Ort_Comp M};
  0.(DualSp X) in Ort_Comp M by RLSUB_1:17; then
A1P: ||.x-0.(DualSp X).|| in B;
  B c= REAL
  proof let r be object;
    assume r in B; then
    ex m be Point of DualSp X st r= ||.x-m.|| & m in Ort_Comp M;
    hence r in REAL;
  end; then
  reconsider B as real-membered set;
  B is bounded_below
  proof
    reconsider r0 = 0 as Real;
    take r0;
    let r be ExtReal;
    assume r in B; then
    ex m be Point of DualSp X st r= ||.x-m.|| & m in Ort_Comp M;
    hence r0 <= r;
  end;
  hence thesis by A1P;
end;
