reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th23:
  for X be non empty closed_interval Subset of REAL,
      Y be RealNormSpace,
      seq be sequence of R_NormSpace_of_ContinuousFunctions(X,Y)
  st Y is complete &
    seq is Cauchy_sequence_by_Norm holds seq is convergent
proof
  let X be non empty closed_interval Subset of REAL;
  let Y be RealNormSpace;
  let vseq be sequence of R_NormSpace_of_ContinuousFunctions(X,Y);
  assume A1: Y is complete & vseq is Cauchy_sequence_by_Norm;
  rng vseq c= BoundedFunctions(X,Y) by XBOOLE_1:1; then
  reconsider vseq1=vseq as sequence of
    R_NormSpace_of_BoundedFunctions(X,Y) by FUNCT_2:6;
  now let e be Real such that
  A2: e >0;
    consider k be Nat such that
  A3: for n,m be Nat st n >= k & m >= k holds
    ||. vseq.n - vseq.m .|| < e by A1,A2,RSSPACE3:8;
    take k;
    now let n,m be Nat;
      assume n >= k & m >= k; then
      ||. vseq.n - vseq.m .|| < e by A3;
      hence ||. vseq1.n - vseq1.m .|| < e by Th17,Th21;
    end;
    hence for n,m be Nat st n >= k & m >= k holds
          ||. vseq1.n - vseq1.m .|| < e;
  end; then
  A4:vseq1 is convergent by A1,LOPBAN_1:def 15,RSSPACE3:8;
    reconsider Z = ContinuousFunctions(X,Y) as Subset of
      R_NormSpace_of_BoundedFunctions(X,Y);
    rng vseq c= ContinuousFunctions(X,Y); then
    reconsider tv=lim vseq1 as Point of
    R_NormSpace_of_ContinuousFunctions(X,Y) by A4,NFCONT_1:def 3,Th22;
    for e be Real st e > 0 ex k be Nat
      st for n be Nat st
      n >= k holds ||.vseq.n - tv.|| < e
    proof
      let e be Real;
      assume e > 0; then
      consider k be Nat such that
    A5: for n be Nat st n >= k holds
          ||.vseq1.n - lim vseq1.|| < e by A4,NORMSP_1:def 7;
      take k;
      now let n be Nat;
        assume n >= k; then
        ||.vseq1.n-lim vseq1.|| < e by A5;
        hence ||.vseq.n-tv.|| < e by Th17,Th21;
      end;
      hence for n be Nat st n >= k holds ||.vseq.n - tv.|| < e;
    end;
  hence thesis by NORMSP_1:def 6;
end;
