reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem
  f is_differentiable_on Y implies Y is open
  proof
    assume
A1: f is_differentiable_on Y;
    now
      let x0 be Element of REAL;
      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 st x in N
      holds (f|Y)/.x-(f|Y)/.x0=L/.(x-x0)+R/.(x-x0);
      take N;
      thus N c= Y by A2,XBOOLE_1:1;
    end;
    hence thesis by RCOMP_1:20;
  end;
