reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;

theorem Th6:
  f is_differentiable_on Y implies Y is open
proof
  assume A1: f is_differentiable_on Y;
  reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
A2: Y c= dom g by A1;
    now
      let x;
      assume x in Y;
      then f|Y is_differentiable_in x by A1;
      hence g|Y is_differentiable_in x;
    end;
    then g is_differentiable_on Y by A2,NDIFF_3:def 5;
    hence Y is open by NDIFF_3:11;
end;
