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;
reserve n for non zero Element of NAT,
  s, t, o for Point of TOP-REAL n;

theorem Th9:
  for x being Point of Tdisk(o,r), f being Function of Tdisk(o,r),
Tdisk(o,r) st not x is_a_fixpoint_of f & x is Point of Tcircle(o,r) holds HC(x,
  f) = x
proof
  let x be Point of Tdisk(o,r);
  let f be Function of Tdisk(o,r), Tdisk(o,r) such that
A1: not x is_a_fixpoint_of f and
A2: x is Point of Tcircle(o,r);
A3: x <> f.x by A1;
A4: the carrier of Tcircle(o,r) = Sphere(o,r) by TOPREALB:9;
  consider y, z being Point of TOP-REAL n such that
A5: y = x and
A6: z = f.x & HC(x,f) = HC(z,y,o,r) by A1,Def4;
  x in halfline(z,y) by A5,TOPREAL9:28;
  then x in halfline(z,y) /\ Sphere(o,r) by A2,A4,XBOOLE_0:def 4;
  hence thesis by A3,A5,A6,Def3;
end;
