reserve n for Element of NAT,
  a, r for Real,
  x for Point of TOP-REAL n;
reserve n for Element of NAT,
  r for non negative Real,
  s, t, x for Point of TOP-REAL n;

theorem Th5:
  s <> t & s in the carrier of Tcircle(x,r) & t is Point of Tdisk(x
,r) implies ex e being Point of Tcircle(x,r) st e <> s & {s,e} = halfline(s,t)
  /\ Sphere(x,r)
proof
  assume
A1: s <> t & s in the carrier of Tcircle(x,r) & t is Point of Tdisk(x,r);
  reconsider S = 1/2*s + 1/2*t, T = t, X = x as Element of REAL n by EUCLID:22;
A2: the carrier of Tcircle(x,r) = Sphere(x,r) by TOPREALB:9;
  set a = (-(2*|(t-(1/2*s + 1/2*t),1/2*s + 1/2*t-x)|) + sqrt delta (Sum sqr (T
  -S), 2 * |(t-(1/2*s + 1/2*t),1/2*s + 1/2*t-x)|, Sum sqr (S-X) - r^2)) / (2 *
  Sum sqr (T-S));
  the carrier of Tdisk(x,r) = cl_Ball(x,r) by Th3;
  then consider e1 being Point of TOP-REAL n such that
A3: e1 <> s and
A4: {s,e1} = halfline(s,t) /\ Sphere(x,r) and
  t in Sphere(x,r) implies e1 = t and
  not t in Sphere(x,r) & a = (-(2*|(t-(1/2*s + 1/2*t),1/2*s + 1/2*t-x )|)
+ sqrt delta (Sum sqr (T-S), 2 * |(t-(1/2*s + 1/2*t),1/2*s + 1/2*t-x)|, Sum sqr
  (S-X) - r^2)) / (2 * Sum sqr (T-S)) implies e1 = (1-a)*(1/2*s + 1/2*t) + a* t
  by A1,A2,TOPREAL9:38;
  e1 in {s,e1} by TARSKI:def 2;
  then e1 in Sphere(x,r) by A4,XBOOLE_0:def 4;
  hence thesis by A2,A3,A4;
end;
