 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th27:
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),
    H be non empty Subset of
    (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T))
  st G = H
  holds
  Cl(H) is sequentially_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),
    H be non empty Subset of
    (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T));
assume G = H; then
(MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T))
 | H is totally_bounded
iff
 G is equibounded & G is equicontinuous by Th26, A1, A2, A3;
hence thesis by ASCOLI:11, A2;
end;
