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 X & Z c= X implies f is_differentiable_on Z
proof
  assume that
A1: f is_differentiable_on X and
A2: Z c= X;
A3: f|X is differentiable by A1;
  X c= dom f by A1;
  hence Z c= dom f by A2;
  let x0;
  assume
A4: x0 in dom (f|Z);
  then
A5: x0 in dom f by RELAT_1:57;
A6: x0 in Z by A4,RELAT_1:57;
  then consider N1 being Neighbourhood of x0 such that
A7: N1 c= Z by Th9;
  x0 in dom (f|X) by A2,A5,A6,RELAT_1:57;
  then f|X is_differentiable_in x0 by A3;
  then consider N being Neighbourhood of x0 such that
A8: N c= dom(f|X) and
A9: ex L,R st for x st x in N holds (f|X)/.x-(f|X)/.x0 = L/.(x-x0)+R/.(
  x-x0);
  consider N2 being Neighbourhood of x0 such that
A10: N2 c= N and
A11: N2 c= N1 by Lm1;
A12: N2 c= Z by A7,A11;
  take N2;
  dom(f|X) = dom f/\X by RELAT_1:61;
  then dom(f|X) c= dom f by XBOOLE_1:17;
  then N c= dom f by A8;
  then N2 c= dom f by A10;
  then N2 c= dom f/\Z by A12,XBOOLE_1:19;
  hence
A13: N2 c= dom(f|Z) by RELAT_1:61;
  consider L,R such that
A14: for x st x in N holds (f|X)/.x-(f|X)/.x0 = L/.(x-x0)+R/.(x-x0) by A9;
  take L,R;
  let x;
  assume
A15: x in N2;
  then (f|X)/.x-(f|X)/.x0 = L/.(x-x0)+R/.(x-x0) by A10,A14;
  then
A16: (f|X)/.x-f/.x0 = L/.(x-x0)+R/.(x-x0) by A2,A5,A6,PARTFUN2:17;
  x in N by A10,A15;
  then f/.x-f/.x0 = L/.(x-x0)+R/.(x-x0) by A8,A16,PARTFUN2:15;
  then f/.x-(f|Z)/.x0 = L/.(x-x0)+R/.(x-x0) by A5,A6,PARTFUN2:17;
  hence thesis by A13,A15,PARTFUN2:15;
end;
