
theorem Th4:
  for m be non zero Element of NAT, k be Element of NAT,
      X be non empty Subset of REAL m, f,g be PartFunc of REAL m,REAL
        st f is_continuously_differentiable_up_to_order k,X &
           g is_continuously_differentiable_up_to_order k,X & X is open
    holds
      f+g 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,
      f,g be PartFunc of REAL m,REAL;
  assume
A1: f is_continuously_differentiable_up_to_order k,X
      & g is_continuously_differentiable_up_to_order k,X & X is open;
A2:dom (f+g) = dom f /\ dom g by VALUED_1:def 1;
  for I be non empty FinSequence of NAT st len I <= k & rng I c= Seg m
    holds (f+g)`partial|(X,I) is_continuous_on X
  proof
    let I be non empty FinSequence of NAT;
    assume
A3:   len I <= k & rng I c= Seg m;
A4: f is_partial_differentiable_on X,I by A3,PDIFF_9:def 11,A1;
A5: g is_partial_differentiable_on X,I by A3,A1,PDIFF_9:def 11;
    reconsider f0 = f`partial|(X,I) as PartFunc of REAL m,REAL;
    reconsider g0 = g`partial|(X,I) as PartFunc of REAL m,REAL;
A6: X = dom f0 by Th1,A3,PDIFF_9:def 11,A1;
A7: X = dom g0 by Th1,A3,A1,PDIFF_9:def 11;
A8:f0 is_continuous_on X by A3,A1;
A9:g0 is_continuous_on X by A3,A1;
    f0+g0 is_continuous_on X by A8,A9,A6,A7,PDIFF_9:46;
    hence
      (f+g)`partial|(X,I) is_continuous_on X by A1,A3,A4,A5,PDIFF_9:75;
  end;
  hence thesis by A2,A1,PDIFF_9:85,XBOOLE_1:19;
end;
