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 Th11:
  for o being Point of TOP-REAL n, 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 (BR-map(f)).x = x
proof
  let o be Point of TOP-REAL n;
  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)& x is Point of Tcircle(o,r);
  thus (BR-map(f)).x = HC(x,f) by Def5
    .= x by A1,Th9;
end;
