reserve y for object, X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1 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 f,f1,f2 for PartFunc of REAL,REAL;
reserve h for non-zero 0-convergent Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

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;
