reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;
reserve V for RealLinearSpace,
        p,q,x for Element of V;
reserve p, q, x for Point of TOP-REAL n;
reserve s, t for Point of TOP-REAL 2;

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;
