 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 Th14:
  for M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace
   st S = TopSpaceMetr(M) & T is complete holds
  for G be Subset of Funcs(the carrier of M, the carrier of T),
      H be non empty Subset of
    MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T)
  st G = H &
  (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T))
 | H is totally_bounded holds
( for x be Point of S,
    Hx be non empty Subset of MetricSpaceNorm T
       st Hx = {f.x where f is Function of S,T :f in H }
     holds (MetricSpaceNorm T) | Hx is totally_bounded )
& G is equicontinuous
proof
  let M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace;
  assume A1: S = TopSpaceMetr(M) & T is complete;
  let G be Subset of Funcs(the carrier of M, the carrier of T),
      H be non empty Subset of
      MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T);
  assume A2: G = H;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  set MZH = (MetricSpaceNorm Z) | H;
A3:the carrier of MZH = H by TOPMETR:def 2;
  assume A4: (MetricSpaceNorm Z) | H is totally_bounded;
  thus for x be Point of S,
    Hx be non empty Subset of MetricSpaceNorm T
       st Hx = {f.x where f is Function of S,T :f in H }
     holds (MetricSpaceNorm T) | Hx is totally_bounded
proof
  let x be Point of S,
  Hx be non empty Subset of MetricSpaceNorm T;
  assume A5:Hx = {f.x where f is Function of S,T :f in H };
  set MTHx = (MetricSpaceNorm T) | Hx;
  let e be Real;
  assume 0 < e; then
  consider L being Subset-Family of MZH such that
A6: L is finite & the carrier of MZH = union L
   & for C being Subset of MZH st C in L holds
     ex w being Element of MZH st C = Ball (w,e) by A4;
  defpred P1[object,object] means
   ex w be Point of MZH st $2 = w & $1 =Ball(w,e);
A7: for D be object st D in L
     ex w be object st w in the carrier of MZH & P1[D,w]
  proof
    let D be object;
    assume A8: D in L; then
    reconsider D0=D as Subset of MZH;
    consider w being Element of MZH such that A9: D0 = Ball (w,e) by A6,A8;
    take w;
    thus w in the carrier of MZH & P1[D,w] by A9;
  end;
  consider U being Function of L,the carrier of MZH such that
  A10: for D being object
         st D in L
        holds P1[D,U.D] from FUNCT_2:sch 1(A7);
A11:for D being object st D in L holds D = Ball(U/.D,e)
proof
  let D be object;
  assume A12: D in L; then
A13:  ex x0 be Point of MZH st U.D = x0 & D =Ball(x0,e) by A10;
  dom U = L by FUNCT_2:def 1;
  hence D = Ball(U/.D,e) by A13,A12,PARTFUN1:def 6;
end;
defpred Q1[object,object] means
   ex w be Function of S,T,
      p be Point of MTHx st
    $1 = w & p= w.x & $2 =Ball(p,e);
A14: for f be object st f in (the carrier of MZH)
     ex B be object st B in bool the carrier of MTHx & Q1[f,B]
  proof
    let f be object;
    assume
    A15: f in the carrier of MZH; then
    f in Z by A3; then
     ex g be Function of the carrier of S, the carrier of T
   st f=g & g is continuous; then
   reconsider g = f as Function of S,T;
    g.x in Hx by A15,A5,A3; then
   reconsider p=g.x as Point of MTHx by TOPMETR:def 2;
   take B = Ball(p,e);
   thus thesis;
end;
consider NF being Function of the carrier of MZH,
bool the carrier of MTHx such that
  A16: for D being object
         st D in the carrier of MZH holds Q1[D,NF.D] from FUNCT_2:sch 1(A14);
A17: dom U = L by FUNCT_2:def 1;
set Le = NF .:(rng U);
reconsider Le as Subset-Family of MTHx;
take Le;
thus Le is finite by A6;
for t be object holds t in the carrier of MTHx iff t in union Le
proof
  let t0 be object;
  hereby assume A18: t0 in the carrier of MTHx; then
   A19:t0 in Hx by TOPMETR:def 2;
   reconsider t = t0 as Point of MTHx by A18;
   consider f be Function of S,T such that
   A20:t=f.x & f in H by A5,A19;
   consider K be set such that
   A21: f in K & K in L by TARSKI:def 4,A6,A3,A20;
   U/.K =U.K by PARTFUN1:def 6,A21,A17; then
   A22:U/.K in rng U by FUNCT_1:def 3,A17,A21;
   consider g be Function of S,T,
            p be Point of MTHx such that
   A23:    (U/.K) = g & p= g.x & NF.(U/.K) =Ball(p,e) by A16;
   A24: f in Ball(U/.K,e) by A21,A11;
   reconsider f0= f as Point of MZH by A21;
   A25: dist(f0,U/.K) < e by A24,METRIC_1:11;
  reconsider f1=f0 as Point of MetricSpaceNorm Z
     by TOPMETR:def 1,TARSKI:def 3;
  reconsider g1=U/.K as Point of MetricSpaceNorm Z
     by TOPMETR:def 1,TARSKI:def 3;
 A26: dist(f1,g1) < e by A25,TOPMETR:def 1;
  reconsider f2=f1,g2=g1 as Point of Z;
 A27: ||.f2-g2.|| < e by NORMSP_2:def 1,A26;
g2-f2 in ContinuousFunctions(S,T); then
 ex f be Function of the carrier of S,the carrier of T
  st g2-f2 =f & f is continuous; then
  reconsider fg=g2-f2 as Function of S,T;
  ||.fg.x.|| <= ||.g2-f2.|| by C0SP3:37; then
  ||.fg.x.|| <= ||.f2-g2.|| by NORMSP_1:7; then
  ||.g.x - f.x .|| <= ||.f2-g2.|| by A23,C0SP3:48; then
A30: ||.g/.x-f/.x.|| < e by A27,XXREAL_0:2;
  A31: NF.(U/.K) in Le by A22,FUNCT_2:35;
  reconsider tt=t as Point of MetricSpaceNorm T by A20;
  reconsider pp = p as Point of MetricSpaceNorm T
    by TOPMETR:def 1,TARSKI:def 3;
  reconsider p1 = pp as Point of T;
  dist(pp,tt) < e by NORMSP_2:def 1,A23,A20,A30; then
  dist(p,t) < e by TOPMETR:def 1; then
  f.x in { y where y is Point of MTHx : dist(p,y) < e } by A20; then
  f.x in NF.(U/.K) by A23,METRIC_1:def 14;
  hence t0 in union Le by TARSKI:def 4,A31,A20;
end;
 assume t0 in union Le;
 hence t0 in the carrier of MTHx;
end;
hence the carrier of MTHx = union Le by TARSKI:2;
thus for C being Subset of MTHx st C in Le holds
    ex p being Element of MTHx st C = Ball (p,e)
proof
  let C be Subset of MTHx;
  assume C in Le; then
  consider t be object such that
  A33: t in dom NF & t in rng U & C = NF.t by FUNCT_1:def 6;
  consider w be Function of S,T,
           p be Point of MTHx such that
  A34:  t = w & p= w.x & NF.t=Ball(p,e) by A33,A16;
  take p;
  thus C = Ball(p,e) by A34,A33;
end;
end;
thus G is equicontinuous by Th13,A1,A2,A4;
end;
