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
  for S, T, Y being Element of REAL 2 st S = 1/2*s + 1/2*t & T = t & Y =
|[a,b]| & s <> t & s in circle(a,b,r) & t in closed_inside_of_circle(a,b,r) ex
e being Point of TOP-REAL 2 st e <> s & {s,e} = halfline(s,t) /\ circle(a,b,r)
& (t in Sphere(|[a,b]|,r) implies e = t) & (not t in Sphere(|[a,b]|,r) & w = (-
(2*|(t-(1/2*s + 1/2*t),1/2*s + 1/2*t-|[a,b]|)|) + sqrt delta (Sum sqr (T-S), 2
* |(t-(1/2*s + 1/2*t),1/2*s + 1/2*t-|[a,b]|)|, Sum sqr (S-Y) - r^2)) / (2 * Sum
  sqr (T-S)) implies e = (1-w)*(1/2*s + 1/2*t) + w*t)
proof
  reconsider G = |[a,b]| as Point of Euclid 2 by TOPREAL3:8;
A1: cl_Ball(G,r) = closed_inside_of_circle(a,b,r) & cl_Ball(G,r) = cl_Ball(
  |[a,b ]|,r) by Th12,Th45;
  Sphere(G,r) = circle(a,b,r) by Th47;
  then
A2: Sphere(|[a,b]|,r) = circle(a,b,r) by Th13;
  let S, T, Y be Element of REAL 2;
  assume S = 1/2*s + 1/2*t & T = t & Y = |[a,b]| & s <> t & s in circle(a,b,r
  ) & t in closed_inside_of_circle(a,b,r);
  hence thesis by A1,A2,Th36;
end;
