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;

theorem
  for X being Subset of COMPLEX st X is open for z0 be Complex st z0 in
  X holds ex g be Real st {y where y is Complex : |.y-z0.| < g} c= X
proof
  let X be Subset of COMPLEX such that
A1: X is open;
  let z0 be Complex;
  assume z0 in X;
  then consider N be Neighbourhood of z0 such that
A2: N c= X by A1,Th9;
  consider g be Real such that
  0 < g and
A3: {y where y is Complex : |.y-z0.| < g} c= N by Def5;
  take g;
  thus thesis by A2,A3;
end;
