
theorem CLTh39:
  for V be RealNormSpace, V1 be Subset of V
  holds ClNLin(V1) is RealNormSpace
  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)
      & l = NORMSTR(# Cl(Z),
                      Zero_(Cl(Z), V),
                      Add_(Cl(Z), V),
                      Mult_(Cl(Z), V),
                      Norm_(Cl(Z),V) #) by defClN;
    reconsider W = the RLSStruct of l as Subspace of V by A1,RSSPACE11;
    W is RealLinearSpace;
    hence thesis by RSSPACE3:2;
  end;
