reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;

theorem
  for A being Subset of COMPLEX n holds A is closed iff A` is open
proof
  let A be Subset of COMPLEX n;
  thus A is closed implies A` is open
  proof
    assume
A1: for x st for r st r > 0 ex z st |.z.| < r & x + z in A holds x in A;
    let x;
    assume x in A`;
    then not x in A by XBOOLE_0:def 5;
    then consider r such that
A2: r > 0 and
A3: for z st |.z.| < r holds not x + z in A by A1;
    take r;
    thus 0 < r by A2;
    let z;
    assume |.z.| < r;
    hence thesis by A3,SUBSET_1:29;
  end;
  assume
A4: for x st x in A` ex r st 0 < r & for z st |.z.| < r holds x + z in A`;
  let x such that
A5: for r st r > 0 ex z st |.z.| < r & x + z in A;
  now
    let r;
    assume r > 0;
    then consider z such that
A6: |.z.| < r & x + z in A by A5;
    take z;
    thus |.z.| < r & not x + z in A` by A6,XBOOLE_0:def 5;
  end;
  then not x in A` by A4;
  hence thesis by SUBSET_1:29;
end;
