reserve n for Element of NAT,
  i for Integer,
  a, b, r for Real,
  x for Point of TOP-REAL n;

theorem
  for p, q being Point of Tunit_circle(2) holds Topen_unit_circle(p),
  Topen_unit_circle(q) are_homeomorphic
proof
  set D = Sphere(0.TOP-REAL 2,p1);
  let p, q be Point of Tunit_circle(2);
  set P = Topen_unit_circle(p);
  set Q = Topen_unit_circle(q);
  reconsider p2 = p, q2 = q as Point of TOP-REAL 2 by PRE_TOPC:25;
A1: D\{q} c= D by XBOOLE_1:36;
  D\{p} c= D by XBOOLE_1:36;
  then reconsider
  A = D\{p}, B = D\{q} as Subset of Tcircle(0.TOP-REAL 2,1) by A1,Th9;
A2: Q = Tcircle(0.TOP-REAL 2,1) | B by Lm13,Th22,EUCLID:54
    .= (TOP-REAL 2) | (D \ {q2}) by GOBOARD9:2;
  P = Tcircle(0.TOP-REAL 2,1) | A by Lm13,Th22,EUCLID:54
    .= (TOP-REAL 2) | (D \ {p2}) by GOBOARD9:2;
  hence thesis by A2,Lm13,BORSUK_4:53,EUCLID:54;
end;
