
theorem LM76A:
  for V be RealBanachSpace, V1 be SubRealNormSpace of V
  st ex CV1 be Subset of V st CV1 = the carrier of V1 & CV1 is closed
  holds V1 is RealBanachSpace
  proof
    let V be RealBanachSpace, V1 be SubRealNormSpace of V;
    given CV1 be Subset of V such that
    A1: CV1 = the carrier of V1 & CV1 is closed;
    for seq be sequence of V1 st seq is Cauchy_sequence_by_Norm
    holds seq is convergent
    proof
      let seq be sequence of V1;
      assume
      A2: seq is Cauchy_sequence_by_Norm;
      the carrier of V1 c= the carrier of V by DUALSP01:def 16; then
      reconsider seq1 = seq as sequence of V by FUNCT_2:7;
      for r be Real st r > 0 ex k be Nat st for n, m be Nat st n >= k & m >= k
      holds ||.(seq1.n) - (seq1.m).|| < r
      proof
        let r be Real;
        assume r > 0; then
        consider k be Nat such that
        A3: for n, m be Nat st n >= k & m >= k
        holds ||.(seq.n) - (seq.m).|| < r by A2,RSSPACE3:8;
        take k;
        let n, m be Nat;
        assume n >= k & m >= k; then
        A4: ||.(seq.n) - (seq.m).|| < r by A3;
        - (seq.m) = (-1) * seq.m by RLVECT_1:16
                 .= (-1) * seq1.m by SUBTH0
                 .= - (seq1.m) by RLVECT_1:16; then
        (seq.n) - (seq.m) = (seq1.n) - (seq1.m) by SUBTH0;
        hence ||.(seq1.n) - (seq1.m).|| < r by A4,SUBTH0;
      end; then
      A6: seq1 is convergent by LOPBAN_1:def 15,RSSPACE3:8;
      rng seq c= CV1 by A1; then
      reconsider s = lim seq1 as Point of V1 by A1,A6;
      for r be Real st 0 < r ex m be Nat st for n be Nat st m <= n
      holds ||.(seq.n) - s.|| < r
      proof
        let r be Real;
        assume r > 0; then
        consider m be Nat such that
        A7: for n be Nat st m <= n holds ||.(seq1.n) - lim seq1.|| < r
        by A6,NORMSP_1:def 7;
        take m;
        let n be Nat;
        assume m <= n; then
        A8: ||.(seq1.n) - lim seq1.|| < r by A7;
        - (lim seq1) = (-1) * lim seq1 by RLVECT_1:16
                        .= (-1) * s by SUBTH0
                        .= - s by RLVECT_1:16; then
        (seq1.n)  - (lim seq1) = (seq.n) - s by SUBTH0;
        hence ||.(seq.n) - s.|| < r by A8,SUBTH0;
      end;
      hence seq is convergent;
    end;
    hence thesis by LOPBAN_1:def 15;
  end;
