theorem
  inside_of_circle(a,b,r) c= closed_inside_of_circle(a,b,r)
proof
  let x be object;
  assume
A1: x in inside_of_circle(a,b,r);
  then reconsider x as Point of TOP-REAL 2;
  |. x - |[a,b]| .| < r by A1,Th43;
  hence thesis by Th42;
end;
