 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th28:
for M be non empty MetrSpace, S be non empty compact TopSpace,
    T be NormedLinearTopSpace
  st S = TopSpaceMetr(M) & T is complete & T is finite-dimensional
   & dim (T) <> 0
  holds
for G be Subset of Funcs(the carrier of M, the carrier of T),
    F be non empty Subset of R_NormSpace_of_ContinuousFunctions(S,T)
  st G = F
  holds
    Cl(F) is compact
  iff
    G is equibounded & G is equicontinuous
proof
let M be non empty MetrSpace, S be non empty compact TopSpace,
    T be NormedLinearTopSpace;
assume that
A1: S = TopSpaceMetr(M) and
A2: T is complete and
A3: T is finite-dimensional & dim (T) <> 0;
let G be Subset of Funcs(the carrier of M, the carrier of T),
    F be non empty Subset of
    (R_NormSpace_of_ContinuousFunctions(S,T));
assume A4: G = F;
reconsider H = F as non empty Subset of
          (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T));
A5: Cl(H) = Cl(F) by ASCOLI:1;
Cl(H) is sequentially_compact
iff
G is equibounded & G is equicontinuous by Th27, A1, A2, A3, A4;
hence thesis by TOPMETR4:18, A5;
end;
