theorem Th19: :: ASCOLI2:19

for M being non empty MetrSpace
for S being non empty compact TopSpace
for T being non empty MetrSpace
for F being non empty Subset of (MetricSpace_of_ContinuousFunctions (S,T))
for G being Subset of (Funcs ( the carrier of M, the carrier of T)) st S = TopSpaceMetr M & T is complete & G = F holds
( (MetricSpace_of_ContinuousFunctions (S,T)) | (Cl F) is compact iff ( G is equicontinuous & ( for x being Point of S
for Fx being 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 ) ) )