
theorem
  for V being RealUnitarySpace, u,v,w being Point of V,
      r,p being Real
st v in Ball(u,r) /\ Ball(w,p)
  ex q being Real st Ball(v,q) c= Ball(u,r)
  & Ball(v,q) c= Ball(w,p)
proof
  let V be RealUnitarySpace;
  let u,v,w be Point of V;
  let r,p being Real;
  assume
A1: v in Ball(u,r) /\ Ball(w,p);
  then v in Ball(u,r) by XBOOLE_0:def 4;
  then consider s being Real such that
  s > 0 and
A2: Ball(v,s) c= Ball(u,r) by Th35;
  v in Ball(w,p) by A1,XBOOLE_0:def 4;
  then consider t being Real such that
  t > 0 and
A3: Ball(v,t) c= Ball(w,p) by Th35;
  take q = min(s,t);
  Ball(v,q) c= Ball(v,s) by Th33,XXREAL_0:17;
  hence Ball(v,q) c= Ball(u,r) by A2;
  Ball(v,q) c= Ball(v,t) by Th33,XXREAL_0:17;
  hence thesis by A3;
end;
