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 Th6:
  for g be Real st 0 < g holds
  {y where y is Complex : |.y-z0.| < g} is Neighbourhood of z0
proof
  let g be Real such that
A1: g > 0;
  set N = {y where y is Complex : |.y-z0.| < g};
A2: N c= COMPLEX
  proof
    let z be object;
    assume z in {y where y is Complex : |.y-z0.| < g};
    then ex y be Complex st z = y & |.y-z0.| < g;
    hence thesis by XCMPLX_0:def 2;
  end;
  thus thesis by A1,A2,Def5;
end;
