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 AA being Subset-Family of COMPLEX n st for A being Subset of
  COMPLEX n st A in AA holds A is open for A being Subset of COMPLEX n st A =
  union AA holds A is open
proof
  let AA be Subset-Family of COMPLEX n such that
A1: for A being Subset of COMPLEX n st A in AA holds A is open;
  let A be Subset of COMPLEX n such that
A2: A = union AA;
  let x;
  assume x in A;
  then consider B being set such that
A3: x in B and
A4: B in AA by A2,TARSKI:def 4;
  reconsider B as Subset of COMPLEX n by A4;
  B is open by A1,A4;
  then consider r such that
A5: 0 < r and
A6: for z st |.z.| < r holds x + z in B by A3;
  take r;
  thus 0 < r by A5;
  let z;
  assume |.z.| < r;
  then x + z in B by A6;
  hence thesis by A2,A4,TARSKI:def 4;
end;
