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 Th13:
  for X be Subset of COMPLEX, z0 be Element of COMPLEX, r be
  Real st X = {y where y is Complex : |.y-z0.| < r} holds X is open
proof
  let X be Subset of COMPLEX, z0 be Element of COMPLEX, r be Real;
  reconsider X0 = X as Subset of COMPLEX;
  assume
A1: X = {y where y is Complex : |.y-z0.| < r};
  for x be Complex st x in X0 ex N be Neighbourhood of x st N c= X0
  proof
    let x be Complex;
    reconsider r1 = (r- |.x-z0.|)/2 as Real;
    set N = {y where y is Complex : |.y-x.| < r1};
    assume x in X0;
    then ex y be Complex st x = y & |.y-z0.| < r by A1;
    then
A2: r - |.x-z0.| > 0 by XREAL_1:50;
    then reconsider N as Neighbourhood of x by Th6;
    r1 < r- |.x-z0.| by A2,XREAL_1:216;
    then
A3: r1+|.x-z0.| < (r- |.x-z0.|)+|.x-z0.| by XREAL_1:8;
    take N;
    let z be object;
    assume z in N;
    then consider y1 be Complex such that
A4: z = y1 and
A5: |.y1-x.| < r1;
    |.y1-x.|+|.x-z0.| < r1+|.x-z0.| by A5,XREAL_1:8;
    then |.y1-z0.| <= |.y1-x.|+|.x-z0.| & |.y1-x.|+|.x-z0.| < r by A3,
COMPLEX1:63,XXREAL_0:2;
    then |.y1-z0.| < r by XXREAL_0:2;
    hence thesis by A1,A4;
  end;
  hence thesis by Th11;
end;
