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;
