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 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 closed
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 s1 be Complex_Sequence st rng s1 c= X0 & s1 is convergent holds lim
  s1 in X0
  proof
    reconsider r as Element of REAL by XREAL_0:def 1;
    set s4 = seq_const r;
    reconsider s2 = NAT --> z0 as Complex_Sequence;
    let s1 be Complex_Sequence;
    assume that
A2: rng s1 c= X0 and
A3: s1 is convergent;
    set s3 = s1-s2;
A4: s2 is convergent by CFCONT_1:26;
    then
A5: lim s3 = lim s1-lim s2 by A3,COMSEQ_2:26;
A6: for n be Element of NAT holds (|.s3.|).n <= r
    proof
      let n be Element of NAT;
      now
        let n be Element of NAT;
        s3.n = s1.n+(-s2).n by VALUED_1:1
          .= s1.n-s2.n by VALUED_1:8;
        hence s3.n = s1.n-z0;
      end;
      then
A7:   s3.n = s1.n-z0;
      dom s1 = NAT by FUNCT_2:def 1;
      then s1.n in rng s1 by FUNCT_1:def 3;
      then s1.n in X0 by A2;
      then ex y be Complex st y = s1.n & |.y-z0.| <= r by A1;
      hence thesis by A7,VALUED_1:18;
    end;
    s3 is convergent by A3,A4;
    then
A8: lim |.s3.| = |.lim s3.| by SEQ_2:27;
    reconsider s3 = s1-s2 as convergent Complex_Sequence by A3,A4;
A9: for n be Nat holds |.s3.|.n <= s4.n
    proof
      let n be Nat;
A10:   n in NAT by ORDINAL1:def 12;
      (|.s3.|).n <= r by A6,A10;
      hence thesis by SEQ_1:57;
    end;
A11: for n be Element of NAT holds lim s2 = z0
    proof
      let n be Element of NAT;
      lim s2 = s2.n by CFCONT_1:28
        .= z0;
      hence thesis;
    end;
    lim s4 = s4.0 by SEQ_4:26
      .= r by SEQ_1:57;
    then lim |.s3.| <= r by A9,SEQ_2:18;
    then |.(lim s1)-z0.| <= r by A11,A5,A8;
    hence thesis by A1;
  end;
  hence thesis;
end;
