theorem
  inside_of_circle(a,b,r) \/ circle(a,b,r) = closed_inside_of_circle(a,b ,r)
proof
  reconsider p = |[a,b]| as Point of Euclid 2 by TOPREAL3:8;
A1: cl_Ball(p,r) = closed_inside_of_circle(a,b,r) by Th45;
  Sphere(p,r) = circle(a,b,r) & Ball(p,r) = inside_of_circle(a,b,r) by Th46
,Th47;
  hence thesis by A1,METRIC_1:16;
end;
