 reserve x,X for set,
         n, m, i for Nat,
         p, q for Point of TOP-REAL n,
         A, B for Subset of TOP-REAL n,
         r, s for Real;
reserve N for non zero Nat,
        u,t for Point of TOP-REAL(N+1);

theorem Th5:
  r + |.p-q.| <= s implies Ball(p,r) c= Ball(q,s)
proof
  assume
A1:  r+|.p-q.| <= s;
  let x be object;
  assume
A2: x in Ball(p,r);
  then reconsider x as Point of TOP-REAL n;
|.x-p.|+|.p-q.| < r+|.p-q.| by A2,TOPREAL9:7,XREAL_1:6;
  then
A3: |.x-p.|+|.p-q.| < s by A1,XXREAL_0:2;
A4: x is Element of REAL n by EUCLID:22;
A5:p is Element of REAL n by EUCLID:22;
  q is Element of REAL n by EUCLID:22;
  then |.x-q.| <= |.x-p.| + |.p-q.| by A4,A5,EUCLID:19;
  then |.x-q.| < s by A3,XXREAL_0:2;
  hence thesis;
end;
