
theorem Th10:
  for X be RealNormSpace, Y be RealBanachSpace, X0 be Subset of Y
    st X is Subspace of Y & the carrier of X = X0
       & the normF of X = (the normF of Y ) | (the carrier of X) &
       X0 is closed holds
     X is complete
proof
  let X be RealNormSpace, Y be RealBanachSpace,
      X0 be Subset of Y;
  assume A1: X is Subspace of Y & the carrier of X = X0 &
       the normF of X = (the normF of Y ) | (the carrier of X)
       & X0 is closed;
now let seq be sequence of X;
   assume A2: seq is Cauchy_sequence_by_Norm;
  rng seq c= the carrier of Y by A1,XBOOLE_1:1; then
  reconsider yseq=seq as sequence of Y by FUNCT_2:6;
  for r be Real st r > 0 ex k
     be Nat st for n, m be Nat st n >= k & m >= k
  holds ||.(yseq.n) - (yseq.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 RSSPACE3:8,A2;
    take k;
    now let n, m be Nat;
      assume A4: n >= k & m >= k;
      (seq.n) - (seq.m) = (yseq.n) - (yseq.m) by A1,RLSUB_1:16;
      then ||.(seq.n) - (seq.m).||
        = ||.(yseq.n) - (yseq.m).|| by FUNCT_1:49,A1;
     hence ||.(yseq.n) - (yseq.m).|| < r by A4,A3;
   end;
  hence thesis;
 end; then
  A5: yseq is convergent by LOPBAN_1:def 15, RSSPACE3:8;
   rng yseq = rng seq;
   then reconsider lyseq=lim yseq as Point of X by A1,A5,NFCONT_1:def 3;
  for r be Real st 0 < r ex m be Nat st for n be Nat
          st m <= n holds ||.(seq.n) - lyseq.|| < r
 proof
  let r be Real;
    assume 0 < r; then
    consider m be Nat such that
A6: for n be Nat
    st m <= n holds ||.(yseq.n) - lim yseq.|| < r by A5,NORMSP_1:def 7;
   take m;
   now let n be Nat;
    assume A7: m <= n;
      (yseq.n) - (lim yseq) = (seq.n) - (lyseq) by A1,RLSUB_1:16;
      then ||.(yseq.n) - (lim yseq).||
        = ||.(seq.n) - (lyseq).|| by FUNCT_1:49,A1;
     hence ||.(seq.n) - (lyseq).|| < r by A7,A6;
   end;
   hence thesis;
  end;
  hence seq is convergent;
end;
hence thesis;
end;
