 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 Th50:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace holds
  for Y be Subset of the carrier of
         R_NormSpace_of_BoundedFunctions(the carrier of S,T) st
  Y = ContinuousFunctions(S,T) holds Y is closed
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let Y be Subset of the carrier of
    R_NormSpace_of_BoundedFunctions(the carrier of S,T);
  assume
A1: Y = ContinuousFunctions(S,T);
  now let seq be sequence of
    R_NormSpace_of_BoundedFunctions(the carrier of S,T);
    assume
A2: rng seq c= Y & seq is convergent;
    reconsider f = lim seq as bounded
          Function of the carrier of S, the carrier of T
            by RSSPACE4:def 5;
A3: dom f = the carrier of S by FUNCT_2:def 1;
    now let z be object;
      assume z in BoundedFunctions (the carrier of S,T); then
      z is Function of the carrier of S, the carrier of T
        by RSSPACE4:def 5;
      hence z in PFuncs(the carrier of S, the carrier of T) by PARTFUN1:45;
    end; then
    BoundedFunctions (the carrier of S, T)
      c= PFuncs(the carrier of S, the carrier of T); then
    reconsider H = seq as Functional_Sequence of the carrier of S,
      the carrier of T by FUNCT_2:7;
A4: for n being Nat holds the carrier of S c= dom (H . n)
    proof
      let n be Nat;
      H.n in rng seq by ORDINAL1:def 12,FUNCT_2:4; then
      H.n in Y by A2; then
      consider f be Function of the carrier of S, the carrier of T such that
A5:   H.n = f & f is continuous by A1;
      thus thesis by A5,FUNCT_2:def 1;
    end;
A6: for p being Real st p > 0 holds ex k being Nat st
      for n being Nat for x be Element of the carrier of S
      st n >= k & x in the carrier of S holds
        ||.(((H . n) /. x) - (f /. x)).|| < p
    proof
      let p be Real;
      assume p > 0;
      then consider k being Nat such that
A7:   for n being Nat st n >= k holds
        ||.((seq . n) - (lim seq)).|| < p by A2, NORMSP_1:def 7;
      reconsider k as Element of NAT by ORDINAL1:def 12;
      take k;
      hereby
        let n be Nat;
        let x be Element of S;
        assume n >= k & x in the carrier of S; then
A8:     ||.((seq . n) - (lim seq)).|| < p by A7;
        reconsider g = (seq . n) - (lim seq) as bounded Function of
          the carrier of S, the carrier of T by RSSPACE4:def 5;
        reconsider s = seq . n as bounded Function of
          the carrier of S, the carrier of T by RSSPACE4:def 5;
        reconsider x0 = x as Element of the carrier of S;
A9:     g . x0 = (s . x0) - (f . x0) by RSSPACE4:24;
A10:    ||.(g . x0).|| <= ||.((seq . n) - (lim seq)).|| by RSSPACE4:16;
        H.n in rng seq by ORDINAL1:def 12,FUNCT_2:4; then
        H.n in Y by A2; then
        ex f be Function of the carrier of S, the carrier of T
          st H.n = f & f is continuous by A1; then
        dom (H.n) = the carrier of S by FUNCT_2:def 1; then
        (H.n).x = (H.n)/.x by PARTFUN1:def 6;
        hence ||.(((H . n) /. x) - (f /. x)).|| < p by A9,A10,A8,XXREAL_0:2;
      end;
    end;
A12:for x be Element of the carrier of S
      st x in the carrier of S holds
    for p be Real st p > 0 ex k be Nat st for n be Nat st n>=k holds
      ||.(H.n)/.x  - f/.x.|| < p
    proof
      let x be Element of the carrier of S;
      assume x in the carrier of S;
      let p be Real;
      assume p > 0; then
      consider k being Nat such that
A13:  for n being Nat
      for x be Element of the carrier of S
      st n >= k & x in the carrier of S holds
      ||.(((H . n) /. x) - (f /. x)).|| < p by A6;
      take k;
      let n be Nat;
      assume n>=k;
      hence ||.(((H . n) /. x) - (f /. x)).|| < p by A13;
    end; then
A14:H is_point_conv_on (the carrier of S) by A4,SEQFUNC:def 9,A3;
A15:H is_unif_conv_on (the carrier of S) by A3,A4,SEQFUNC:def 9,A6;
A16:lim(H,the carrier of S) = f by A3,A12,A14,SEQFUNC2:11;
    for n be Nat holds
     ex Hn be Function of S,T st Hn = (H.n) & Hn is continuous
    proof
      let n be Nat;
      H.n in rng seq by ORDINAL1:def 12,FUNCT_2:4; then
      H.n in Y by A2;then
      consider f be Function of the carrier of S, the carrier of T such that
A17:  H.n = f & f is continuous by A1;
      thus thesis by A17;
    end; then
    f is continuous by Th49,A15,A16;
    hence lim seq in Y by A1;
  end;
  hence thesis;
end;
