
theorem
  for V be RealNormSpace,
      V1 be SubRealNormSpace of V,
      CV1 be Subset of V
  st V1 is complete & CV1 = the carrier of V1
  holds CV1 is closed
  proof
    let V be RealNormSpace,
        V1 be SubRealNormSpace of V,
        CV1 be Subset of V;
    assume
    A1: V1 is complete & CV1 = the carrier of V1;
    for s1 be sequence of V st rng s1 c= CV1 & s1 is convergent
    holds lim s1 in CV1
    proof
      let s1 be sequence of V;
      assume
      A2: rng s1 c= CV1 & s1 is convergent; then
      reconsider s2 = s1 as sequence of V1 by A1,FUNCT_2:6;
      for s be Real st 0 < s ex n be Nat st
      for m be Nat st n <= m holds ||.s2.m -s2.n.|| < s
      proof
        let s be Real;
        assume 0 < s; then
        consider n be Nat such that
        A3: for m be Nat st n <= m holds ||.s1.m -s1.n.|| < s
        by A2,LOPBAN_3:4;
        take n;
        let m be Nat;
        assume n <= m; then
        A4: ||.s1.m -s1.n.|| < s by A3;
        - (s1.n) = (-1) * s1.n by RLVECT_1:16
                .= (-1) * s2.n by SUBTH0
                .= - (s2.n) by RLVECT_1:16; then
        (s1.m) - (s1.n) = (s2.m) - (s2.n) by SUBTH0;
        hence ||.(s2.m) - (s2.n).|| < s by A4,SUBTH0;
      end; then
      A6: s2 is convergent by A1,LOPBAN_1:def 15,LOPBAN_3:5;
      the carrier of V1 c= the carrier of V by DUALSP01:def 16; then
      reconsider s0 = lim s2 as Point of V;
      for r be Real st 0 < r ex m be Nat
      st for n be Nat st m <= n holds ||.(s1.n) - s0.|| < r
      proof
        let r be Real;
        assume 0 < r; then
        consider m be Nat such that
        A7: for n be Nat st m <= n holds ||.(s2.n) - lim s2.|| < r
        by A6,NORMSP_1:def 7;
        take m;
        let n be Nat;
        assume m <= n; then
        A8: ||.(s2.n) - lim s2.|| < r by A7;
        - (lim s2) = (-1) * lim s2 by RLVECT_1:16
                  .= (-1) * s0 by SUBTH0
                  .= - s0 by RLVECT_1:16; then
        (s2.n) - (lim s2) = (s1.n) - s0 by SUBTH0;
        hence ||.(s1.n) - s0.|| < r by A8,SUBTH0;
      end; then
      lim s1 = s0 by A2,NORMSP_1:def 7;
      hence lim s1 in CV1 by A1;
    end;
    hence thesis;
  end;
