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,B being Subset of COMPLEX n st A is open & B is open for C
  being Subset of COMPLEX n st C = A /\ B holds C is open
proof
  let A,B be Subset of COMPLEX n such that
A1: A is open and
A2: B is open;
  let C be Subset of COMPLEX n such that
A3: C = A /\ B;
  let x;
  assume
A4: x in C;
  then x in A by A3,XBOOLE_0:def 4;
  then consider r1 such that
A5: 0 < r1 and
A6: for z st |.z.| < r1 holds x + z in A by A1;
  x in B by A3,A4,XBOOLE_0:def 4;
  then consider r2 such that
A7: 0 < r2 and
A8: for z st |.z.| < r2 holds x + z in B by A2;
  take min(r1,r2);
  thus 0 < min(r1,r2) by A5,A7,XXREAL_0:15;
  let z;
  assume
A9: |.z.| < min(r1,r2);
  min(r1,r2) <= r2 by XXREAL_0:17;
  then |.z.| < r2 by A9,XXREAL_0:2;
  then
A10: x + z in B by A8;
  min(r1,r2) <= r1 by XXREAL_0:17;
  then |.z.| < r1 by A9,XXREAL_0:2;
  then x + z in A by A6;
  hence thesis by A3,A10,XBOOLE_0:def 4;
end;
