theorem
  cl_Ball(|[a,b]|,r) = closed_inside_of_circle(a,b,r)
proof
  reconsider e = |[a,b]| as Point of Euclid 2 by TOPREAL3:8;
  thus cl_Ball(|[a,b]|,r) = cl_Ball(e,r) by Th12
    .= closed_inside_of_circle(a,b,r) by Th45;
end;
