
theorem
  for V be RealNormSpace,
      W be SubRealNormSpace of V
  holds W is Subspace of V
  proof
    let V be RealNormSpace,
        W be SubRealNormSpace of V;
    the carrier of W c= the carrier of V
      & 0.W = 0.V
      & the addF of W = (the addF of V) || (the carrier of W)
      & the Mult of W = (the Mult of V) | [:REAL, the carrier of W:]
      & the normF of W = (the normF of V) | (the carrier of W)
      by DUALSP01:def 16;
    hence thesis by RLSUB_1:def 2;
  end;
