
theorem Th3:
  for m be non zero Element of NAT, X be non empty Subset of REAL m,
      f be PartFunc of REAL m,REAL st X is open & X c= dom f
                         & f is_continuously_differentiable_up_to_order 1,X
  holds
    f is_continuous_on X
proof
  let m be non zero Element of NAT,
      X be non empty Subset of REAL m,
      f be PartFunc of REAL m,REAL;
  assume
A1: X is open & X c= dom f & f is_continuously_differentiable_up_to_order 1,X;
  then
A2: f is_differentiable_on X by Th2;
  reconsider g = <>*f as PartFunc of REAL m,REAL 1;
A3: g is_differentiable_on X by A1,A2,PDIFF_9:53;
  the carrier of (REAL-NS m) = REAL m
    & the carrier of (REAL-NS 1) = REAL 1 by REAL_NS1:def 4;
  then
    reconsider h = <>*f as PartFunc of REAL-NS m,REAL-NS 1;
  h is_differentiable_on X by A3,PDIFF_6:30;
  then h is_continuous_on X by NDIFF_1:45;
  then g is_continuous_on X by PDIFF_7:37;
  hence f is_continuous_on X by PDIFF_9:44;
end;
