 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th51:
  for S be non empty compact TopSpace,
      T be NormedLinearTopSpace st T is complete holds
  for seq be sequence of R_NormSpace_of_ContinuousFunctions(S,T)
   st seq is Cauchy_sequence_by_Norm holds seq is convergent
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  assume A1: T is complete;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  let vseq be sequence of Z;
  assume
A2: vseq is Cauchy_sequence_by_Norm;
A3:for x being object st x in ContinuousFunctions(S,T) holds
     x in BoundedFunctions(the carrier of S,T) by Th34;
   rng vseq c= BoundedFunctions(the carrier of S,T) by Th34; then
   reconsider vseq1=vseq as sequence of
     R_NormSpace_of_BoundedFunctions(the carrier of S,T) by FUNCT_2:6;
A4: now let e be Real;
     assume e > 0; then
     consider k be Nat such that
A6:  for n,m be Nat st n >= k & m >= k holds
       ||. vseq.n - vseq.m .|| < e by A2,RSSPACE3:8;
     take k;
     let n,m be Nat;
     assume n >= k & m >= k; then
     ||. vseq.n - vseq.m .|| < e by A6;
     hence ||. vseq1.n - vseq1.m .|| < e by Th47,Th36;
  end; then
A7:vseq1 is Cauchy_sequence_by_Norm by RSSPACE3:8;
A8: vseq1 is convergent by A1,RSSPACE4:25,A4,RSSPACE3:8;
  reconsider Y = ContinuousFunctions(S,T) as Subset of
     R_NormSpace_of_BoundedFunctions(the carrier of S,T) by A3,TARSKI:def 3;
A9:rng vseq c= Y;
  Y is closed by Th50; then
  reconsider tv=lim vseq1 as Point of Z by A1,A9,A7,C0SP2:20;
  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
A10:for n be Nat st n >= k holds
      ||.vseq1.n - lim vseq1.|| < e by A8,NORMSP_1:def 7;
    take k;
    now
      let n be Nat;
      assume n >= k; then
      ||.vseq1.n-lim vseq1.|| < e by A10;
      hence ||.vseq.n-tv.|| < e by Th47,Th36;
    end;
    hence for n be Nat st n >= k holds ||.vseq.n - tv.|| < e;
  end;
  hence thesis;
end;
