theorem
  D = Ball(e,r) & p = e implies N-bound D = p`2 + r
proof
  assume that
A1: D = Ball(e,r) and
A2: p = e;
  r > 0 by A1,TBSP_1:12;
  then
A3: p`2+0 < p`2+r by XREAL_1:6;
  p`2-r < p`2-0 by A1,TBSP_1:12,XREAL_1:15;
  then p`2-r < p`2+r by A3,XXREAL_0:2;
  then
A4: upper_bound ].p`2-r,p`2+r.[ = p`2+r by Th16;
  proj2.:D = ].p`2-r,p`2+r.[ by A1,A2,Th42;
  hence thesis by A4,SPRECT_1:45;
end;
