
theorem Th35:
  for V being RealUnitarySpace, v,u being Point of V, r being Real
  st u in Ball(v,r)
   ex p being Real st p>0 & Ball(u,p) c= Ball(v,r)
proof
  let V be RealUnitarySpace;
  let v,u be Point of V;
  let r be Real;
  assume u in Ball(v,r);
  then
A1: dist(v,u) < r by BHSP_2:41;
  thus thesis
  proof
    set p = r - dist(v,u);
    take p;
    thus p > 0 by A1,XREAL_1:50;
    for w being Point of V holds w in Ball(u,p) implies w in Ball(v,r)
    proof
      let w be Point of V;
      assume w in Ball(u,p);
      then dist(u,w) < r - dist(v,u) by BHSP_2:41;
      then
A2:   dist(v,u) + dist(u,w) < r by XREAL_1:20;
      dist(v,u) + dist(u,w) >= dist(v,w) by BHSP_1:35;
      then dist(v,w) < r by A2,XXREAL_0:2;
      hence thesis by BHSP_2:41;
    end;
    hence thesis;
  end;
end;
