
theorem ThSubSpace:
  for V be RealNormSpace, V1 be Subset of V
  holds NLin(V1) is SubRealNormSpace of V
  proof
    let V be RealNormSpace;
    let V1 be Subset of V;
    set l = NLin(V1);
    A1: the carrier of l c= the carrier of V by RLSUB_1:def 2;
    A2: l is reflexive discerning RealNormSpace-like
      vector-distributive scalar-distributive scalar-associative scalar-unital
      Abelian add-associative right_zeroed right_complementable
      by RSSPACE:15,XTh7;
    A3: 0.l = 0.V by RLSUB_1:def 2;
    A4: the addF of l
         = (the addF of V) || (the carrier of l) by RLSUB_1:def 2;
    A5: the Mult of l
         = (the Mult of V) | [:REAL, the carrier of l:] by RLSUB_1:def 2;
    the normF of l = (the normF of V) | (the carrier of l) by A1,DefNorm;
    hence thesis by A1,A2,A3,A4,A5,DUALSP01:def 16;
  end;
