reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1 for sequence of S;
reserve x0 for Point of S;
reserve h for (0.S)-convergent sequence of S;
reserve c for constant sequence of S;
reserve R,R1,R2 for RestFunc of S,T;
reserve L,L1,L2 for Point of R_NormSpace_of_BoundedLinearOperators(S,T);

theorem
  for f be PartFunc of S,T for Y be Subset of S holds f
  is_differentiable_on Y implies Y is open
proof
  let f be PartFunc of S,T;
  let Y be Subset of S;
  assume
A1: f is_differentiable_on Y;
  now
    let x0 be Point of S;
    assume x0 in Y;
    then f|Y is_differentiable_in x0 by A1;
    then consider N being Neighbourhood of x0 such that
A2: N c= dom(f|Y) and
    ex L,R st for x be Point of S st x in N holds (f|Y)/.x-(f|Y)/.x0=L.( x
    -x0)+R/.(x-x0);
    take N;
    dom(f|Y)=dom f/\Y by RELAT_1:61;
    then dom(f|Y) c=Y by XBOOLE_1:17;
    hence N c= Y by A2;
  end;
  hence thesis by Th4;
end;
