theorem :: ASCOLI:17

for M being non empty MetrSpace
for S being non empty compact TopSpace
for T being NormedLinearTopSpace
for G being Subset of (Funcs ( the carrier of M, the carrier of T))
for H being non empty Subset of (MetricSpaceNorm (R_NormSpace_of_ContinuousFunctions (S,T))) st S = TopSpaceMetr M & T is complete & G = H holds
( Cl H is sequentially_compact iff ( G is equicontinuous & ( for x being Point of S
for Hx being non empty Subset of (MetricSpaceNorm T) st Hx = { (f . x) where f is Function of S,T : f in H } holds
(MetricSpaceNorm T) | (Cl Hx) is compact ) ) )