
theorem Th5:
  for m be non zero Element of NAT, k be Element of NAT,
  X be non empty Subset of REAL m,r be Real,
  f be 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
proof
  let m be non zero Element of NAT, k be Element of NAT,
      X be non empty Subset of REAL m,r be Real,
      f be PartFunc of REAL m,REAL;
  assume
A1: f is_continuously_differentiable_up_to_order k,X & X is open;
  for I be 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
    let I be non empty FinSequence of NAT;
    assume
A2:   len I <= k & rng I c= Seg m;
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;
  end;
  hence thesis by PDIFF_9:86,A1,VALUED_1:def 5;
end;
