 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);
 reserve S,Z for RealNormSpace;
 reserve T for RealBanachSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th6:
for Z be RealNormSpace
   holds
Z is complete iff MetricSpaceNorm Z is complete
proof
  let Z be RealNormSpace;
  set N = MetricSpaceNorm Z;
  hereby assume A1:Z is complete;
    for S2 being sequence of N st S2 is Cauchy holds
    S2 is convergent
  proof
    let S2 be sequence of N;
    assume A2: S2 is Cauchy;
    reconsider seq2 = S2 as sequence of Z;
    for r being Real st r > 0 holds
     ex k being Nat st
     for n, m being Nat st n >= k & m >= k holds
      ||.((seq2 . n) - (seq2 . m)).|| < r
proof
  let r be Real;
  assume r > 0; then
  consider p being Nat such that
   A3:for n, m being Nat st p <= n & p <= m holds
    dist ((S2 . n),(S2 . m)) < r by A2;
  take p;
  let n, m be Nat;
    assume p <= n & p <= m; then
    dist ((S2 . n),(S2 . m)) < r by A3;
    hence ||.((seq2 . n) - (seq2 . m)).|| < r by NORMSP_2:def 1;
end; then
A4: seq2 is convergent by A1,RSSPACE3:8;
reconsider L = lim seq2 as Point of N;
for r being Real st 0 < r holds
  ex m being Nat st
 for n being Nat st m <= n holds
  ||.((seq2 . n) - lim seq2).|| < r by A4,NORMSP_1:def 7;
then
S2 is_convergent_in_metrspace_to L by NORMSP_2:4;
hence S2 is convergent by METRIC_6:9;
end;
hence MetricSpaceNorm Z is complete;
end;
assume A5: MetricSpaceNorm Z is complete;
for seq2 be sequence of Z st seq2 is Cauchy_sequence_by_Norm
holds seq2 is convergent
proof
 let seq2 be sequence of Z;
 assume A6:seq2 is Cauchy_sequence_by_Norm;
 reconsider S2= seq2 as sequence of N;
now let r be Real;
    assume r > 0; then
  consider p being Nat such that
  A7:  for n, m being Nat st p <= n & p <= m holds
   ||.((seq2 . n) - (seq2 . m)).|| < r by A6,RSSPACE3:8;
  take p;
  let n, m be Nat;
  assume p <= n & p <= m; then
  ||.((seq2 . n) - (seq2 . m)).|| < r by A7;
  hence dist ((S2 . n),(S2 . m)) < r by NORMSP_2:def 1;
 end; then
  S2 is Cauchy; then
  S2 is convergent by A5;
hence seq2 is convergent by NORMSP_2:5;
end;
hence Z is complete;
end;
