reserve k, k1, n, n1, m for Nat;
reserve X, y for set;
reserve p for Real;
reserve r for Real;
reserve a, a1, a2, b, b1, b2, x, x0, z, z0 for Complex;
reserve s1, s3, seq, seq1 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f, f1, f2 for PartFunc of COMPLEX,COMPLEX;
reserve Nseq for increasing sequence of NAT;
reserve h for 0-convergent non-zero Complex_Sequence;
reserve c for constant Complex_Sequence;
reserve R, R1, R2 for C_RestFunc;
reserve L, L1, L2 for C_LinearFunc;
reserve Z for open Subset of COMPLEX;

theorem
  f is_differentiable_on Y implies Y is open
proof
  assume
A1: f is_differentiable_on Y;
  then
A2: Y c= dom f;
  now
    let x0 be Complex;
    assume x0 in Y;
    then
A3: x0 in dom (f|Y) by A2,RELAT_1:57;
    f|Y is differentiable by A1;
    then f|Y is_differentiable_in x0 by A3;
    then consider N being Neighbourhood of x0 such that
A4: 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;
    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 A4;
  end;
  hence thesis by Th11;
end;
