let M be non empty MetrSpace; :: thesis: for S being non empty compact TopSpace
for T being NormedLinearTopSpace
for F being non empty Subset of (R_NormSpace_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
( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) )
let S be non empty compact TopSpace; :: thesis: for T being NormedLinearTopSpace
for F being non empty Subset of (R_NormSpace_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
( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) )
let T be NormedLinearTopSpace; :: thesis: for F being non empty Subset of (R_NormSpace_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
( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) )
let F be non empty Subset of (R_NormSpace_of_ContinuousFunctions (S,T)); :: thesis: for G being Subset of (Funcs ( the carrier of M, the carrier of T)) st S = TopSpaceMetr M & T is complete & G = F holds
( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) )
let G be Subset of (Funcs ( the carrier of M, the carrier of T)); :: thesis: ( S = TopSpaceMetr M & T is complete & G = F implies ( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) ) )
assume A1: ( S = TopSpaceMetr M & T is complete ) ; :: thesis: ( not G = F or ( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) ) )
assume G = F ; :: thesis: ( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) )
then ( Cl F is compact iff ( G is equicontinuous & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) ) by Th18, A1;
hence ( Cl F is compact iff ( ( for x being Point of M holds G is_equicontinuous_at x ) & ( for x being Point of S
for Fx being non empty Subset of (MetricSpaceNorm T) st Fx = { (f . x) where f is Function of S,T : f in F } holds
(MetricSpaceNorm T) | (Cl Fx) is compact ) ) ) by Th5, A1; :: thesis: verum