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) misses circle(a,b,r)
proof
  assume not thesis;
  then consider x being object such that
A1: x in inside_of_circle(a,b,r) and
A2: x in circle(a,b,r) by XBOOLE_0:3;
  reconsider x as Point of TOP-REAL 2 by A1;
  |. x - |[a,b]| .| = r by A2,Th41;
  hence thesis by A1,Th43;
end;
