let m be non zero Element of NAT ; :: thesis: for k being Element of NAT

for X being non empty Subset of (REAL m)

for r being Real

for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let k be Element of NAT ; :: thesis: for X being non empty Subset of (REAL m)

for r being Real

for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let X be non empty Subset of (REAL m); :: thesis: for r being Real

for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let r be Real; :: thesis: for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let f be PartFunc of (REAL m),REAL; :: thesis: ( f is_continuously_differentiable_up_to_order k,X & X is open implies r (#) f is_continuously_differentiable_up_to_order k,X )

assume A1: ( f is_continuously_differentiable_up_to_order k,X & X is open ) ; :: thesis: r (#) f is_continuously_differentiable_up_to_order k,X

for I being non empty FinSequence of NAT st len I <= k & rng I c= Seg m holds

(r (#) f) `partial| (X,I) is_continuous_on X

for X being non empty Subset of (REAL m)

for r being Real

for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let k be Element of NAT ; :: thesis: for X being non empty Subset of (REAL m)

for r being Real

for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let X be non empty Subset of (REAL m); :: thesis: for r being Real

for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let r be Real; :: thesis: for f being PartFunc of (REAL m),REAL st f is_continuously_differentiable_up_to_order k,X & X is open holds

r (#) f is_continuously_differentiable_up_to_order k,X

let f be PartFunc of (REAL m),REAL; :: thesis: ( f is_continuously_differentiable_up_to_order k,X & X is open implies r (#) f is_continuously_differentiable_up_to_order k,X )

assume A1: ( f is_continuously_differentiable_up_to_order k,X & X is open ) ; :: thesis: r (#) f is_continuously_differentiable_up_to_order k,X

for I being non empty FinSequence of NAT st len I <= k & rng I c= Seg m holds

(r (#) f) `partial| (X,I) is_continuous_on X

proof

hence
r (#) f is_continuously_differentiable_up_to_order k,X
by PDIFF_9:86, A1, VALUED_1:def 5; :: thesis: verum
let I be non empty FinSequence of NAT ; :: thesis: ( len I <= k & rng I c= Seg m implies (r (#) f) `partial| (X,I) is_continuous_on X )

assume A2: ( len I <= k & rng I c= Seg m ) ; :: thesis: (r (#) f) `partial| (X,I) is_continuous_on X

A3: f is_partial_differentiable_on X,I by A2, PDIFF_9:def 11, A1;

reconsider f0 = f `partial| (X,I) as PartFunc of (REAL m),REAL ;

X = dom f0 by Th1, A2, PDIFF_9:def 11, A1;

then r (#) f0 is_continuous_on X by A2, A1, PDIFF_9:47;

hence (r (#) f) `partial| (X,I) is_continuous_on X by A1, A2, A3, PDIFF_9:79; :: thesis: verum

end;assume A2: ( len I <= k & rng I c= Seg m ) ; :: thesis: (r (#) f) `partial| (X,I) is_continuous_on X

A3: f is_partial_differentiable_on X,I by A2, PDIFF_9:def 11, A1;

reconsider f0 = f `partial| (X,I) as PartFunc of (REAL m),REAL ;

X = dom f0 by Th1, A2, PDIFF_9:def 11, A1;

then r (#) f0 is_continuous_on X by A2, A1, PDIFF_9:47;

hence (r (#) f) `partial| (X,I) is_continuous_on X by A1, A2, A3, PDIFF_9:79; :: thesis: verum