
theorem CLTh40:
  for V be RealNormSpace, V1 be Subset of V
  holds ClNLin(V1) is SubRealNormSpace of V
  proof
    let V be RealNormSpace, V1 be Subset of V;
    set l = ClNLin(V1);
    consider Z be Subset of V such that
    A1: Z = the carrier of Lin(V1)
      & ClNLin(V1) = NORMSTR(# Cl(Z),
                               Zero_(Cl(Z), V),
                               Add_(Cl(Z), V),
                               Mult_(Cl(Z), V),
                               Norm_(Cl(Z),V) #) by defClN;
    reconsider CL = Cl(Z) as Subset of V;
    A3: 0.ClNLin(V1) = 0.V by A1,Cl01,RSSPACE:def 10;
    A4: the addF of l = (the addF of V) || the carrier of l
                        by A1,Cl01,RSSPACE:def 8;
    A5: the Mult of l = (the Mult of V)| [:REAL, the carrier of l:]
                        by A1,Cl01,RSSPACE:def 9;
    the normF of l = (the normF of V) | (the carrier of l) by A1,DefNorm;
    hence thesis by A1,A3,A4,A5,DUALSP01:def 16;
  end;
