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
  A <> {} implies Ball(A,r1) is open
proof
  assume
A1: A <> {};
  let x;
  assume x in Ball(A,r1); then
A2: dist(x,A) < r1 by Th118;
  take r = r1 - dist(x,A);
  thus 0 < r by A2,XREAL_1:50;
  let z;
  assume |.z.| < r; then
A3: |.z.| + dist(x,A) < r + dist(x,A) by XREAL_1:6;
  dist(x + z,A) <= |.z.| + dist(x,A) by A1,Th114;
  then dist(x + z,A) < r + dist(x,A) by A3,XXREAL_0:2;
  hence thesis;
end;
