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 Th7:
  for N being Neighbourhood of z0 holds z0 in N
proof
  let N be Neighbourhood of z0;
  consider g be Real such that
A1: 0 < g and
A2: {z where z is Complex : |.z-z0.| < g} c= N by Def5;
  |.z0-z0.| = 0 by COMPLEX1:44;
  then z0 in {z where z is Complex : |.z-z0.| < g} by A1;
  hence thesis by A2;
end;
