 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
for M be non empty MetrSpace,S be non empty compact TopSpace,
    T be NormedLinearTopSpace,
    U be compact Subset of T,
    F be non empty Subset of
    R_NormSpace_of_ContinuousFunctions(S,T),
    G be Subset of Funcs(the carrier of M, the carrier of T)
st S = TopSpaceMetr(M) & T is complete
    & G = F & for f be Function st f in F holds rng f c= U
  holds
( Cl(F) is compact implies G is equibounded & G is equicontinuous )
    &
( G is equicontinuous implies Cl(F) is compact )
proof
  let M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace,U be compact Subset of T;
  let F be non empty Subset of
        R_NormSpace_of_ContinuousFunctions(S,T),
        G be Subset of Funcs(the carrier of M, the carrier of T);
  assume that
A1:  S = TopSpaceMetr(M) and
A2:  T is complete;
  assume
  A3: G = F & for f be Function st f in F holds rng f c= U;
  reconsider H = F as non empty Subset of
     MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T);
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  (MetricSpaceNorm Z) | H is totally_bounded iff
    Cl(F) is compact by A2,Th12;
  hence Cl(F) is compact implies G is equibounded & G is equicontinuous
    by A2,A1,Th13,A3;
  assume
A5:G is equicontinuous;
   for x be Point of S,
       Fx be non empty Subset of MetricSpaceNorm T
       st Fx = {f.x where f is Function of S,T :f in F } holds
    (MetricSpaceNorm T) | Cl(Fx) is compact
proof
  let x be Point of S,
       Fx be non empty Subset of MetricSpaceNorm T;
   assume
   A6: Fx = {f.x where f is Function of S,T :f in F };
  reconsider V = U as Subset of TopSpaceNorm T;
  A7: V is compact by Th21;
A10:  Fx c= V
  proof
    let y be object;
    assume y in Fx; then
    consider f be Function of S,T such that
    A8:y=f.x & f in F by A6;
    A9: f.x in rng f by FUNCT_2:4;
    rng f c= V by A3,A8;
    hence y in V by A8,A9;
 end;
   consider Gx be Subset of TopSpaceMetr MetricSpaceNorm T such that
A11: Fx = Gx & Cl(Fx) = Cl Gx by Def1;
   reconsider HClx=Cl(Gx) as non empty Subset of MetricSpaceNorm T by A11;
A12:TopSpaceNorm T is T_2 by PCOMPS_1:34;
   Cl(Gx) c= Cl V by A10,A11,PRE_TOPC:19; then
   Cl(Gx) c= V by A12,A7,PRE_TOPC:22; then
   Cl(Gx) is compact by Th21,COMPTS_1:9; then
   HClx is sequentially_compact by TOPMETR4:15;
   hence (MetricSpaceNorm T) | Cl(Fx) is compact by A11,TOPMETR4:14;
end;
hence thesis by Th18,A1,A2,A3,A5;
end;
