for r being Real holds
( inside_of_circle (0,0,r) = { p where p is Point of (TOP-REAL 2) : |.p.| < r } & ( r > 0 implies circle (0,0,r) = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = r } ) & outside_of_circle (0,0,r) = { p3 where p3 is Point of (TOP-REAL 2) : |.p3.| > r } & closed_inside_of_circle (0,0,r) = { q where q is Point of (TOP-REAL 2) : |.q.| <= r } & closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } )