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 in ComplexOpenSets n iff A is open
proof
  let B be Subset of COMPLEX n;
  B in {A where A is Subset of COMPLEX n: A is open} iff ex C being Subset
  of COMPLEX n st B = C & C is open;
  hence thesis;
end;
