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
  Ball(z1,r1) is open
proof
  let x;
  assume x in Ball(z1,r1); then
A1: |.z1 - x.| < r1 by Th109;
  take r = r1 - |.z1 - x.|;
  thus 0 < r by A1,XREAL_1:50;
  let z;
  assume |.z.| < r; then
A2: |.z.| + |.z1 - x.| < r + |.z1 - x.| by XREAL_1:6;
  z1 - x - z = z1 - (x + z) by Th74;
  then |.z1 - (x + z).| <= |.z.| + |.z1 - x.| by Th98;
  then |.z1 - (x + z).| < r + |.z1 - x.| by A2,XXREAL_0:2;
  hence thesis by Th109;
end;
