
theorem
  for M be non empty MetrSpace, S be non empty compact TopSpace,
      T be non empty MetrSpace,
      U be compact Subset of TopSpaceMetr T,
      F be non empty Subset of MetricSpace_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
  MetricSpace_of_ContinuousFunctions(S,T) | Cl(F) is compact iff
    G is equicontinuous
  proof
    let M be non empty MetrSpace,S be non empty compact TopSpace,
        T be non empty MetrSpace,U be compact Subset of TopSpaceMetr T;
    let F be non empty Subset of MetricSpace_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;
    set Z = MetricSpace_of_ContinuousFunctions(S,T);
    Cl(F) is sequentially_compact iff Z | F is totally_bounded by Th14,A2; then
    Z | F is totally_bounded iff Z | Cl(F) is compact by TOPMETR4:14;
    hence Z | Cl(F) is compact implies G is equicontinuous by A1,Th15,A3;
    assume
A5: G is equicontinuous;
    for x be Point of S, Fx be non empty Subset of T
       st Fx = {f.x where f is Function of S,T :f in F } holds
      T | Cl(Fx) is compact
    proof
      let x be Point of S, Fx be non empty Subset of T;
      assume
  A6: Fx = {f.x where f is Function of S,T :f in F };
A7:   Fx c= U
      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= U by A3,A8;
        hence y in U by A8,A9;
      end;
      consider Gx be Subset of TopSpaceMetr T such that
A10:  Fx = Gx & Cl(Fx) = Cl Gx by ASCOLI:def 1;
      reconsider HClx=Cl(Gx) as non empty Subset of T by A10;
A11:  TopSpaceMetr T is T_2 by PCOMPS_1:34;
      Cl Gx c= Cl U by A7,A10,PRE_TOPC:19; then
      Cl Gx c= U by A11,PRE_TOPC:22; then
      Cl Gx is compact by COMPTS_1:9; then
      HClx is sequentially_compact by TOPMETR4:15;
      hence T | Cl Fx is compact by A10,TOPMETR4:14;
    end;
    hence thesis by Th19,A1,A2,A3,A5;
  end;
