reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th67:
  for n being non zero Element of NAT for o, p being Point of TOP-REAL n,
  r being positive Real st p in Ball(o,r)
  holds RotateCircle(o,r,p) is without_fixpoints
proof
  let n be non zero Element of NAT;
  let o, p be Point of TOP-REAL n;
  let r be positive Real;
  assume
A1: p in Ball(o,r);
  set f = RotateCircle(o,r,p);
  let x be object;
  assume
A2: x in dom f;
  set S = Tcircle(o,r);
A3: dom f = the carrier of S by FUNCT_2:def 1;
  consider y being Point of TOP-REAL n such that
A4: x = y and
A5: f.x = HC(y,p,o,r) by A1,A2,Def8;
A6: the carrier of S = Sphere(o,r) by TOPREALB:9;
  Sphere(o,r) c= cl_Ball(o,r) by TOPREAL9:17;
  then
A7: y is Point of Tdisk(o,r) by A2,A3,A4,A6,BROUWER:3;
  Ball(o,r) c= cl_Ball(o,r) by TOPREAL9:16;
  then
A8: p is Point of Tdisk(o,r) by A1,BROUWER:3;
  Ball(o,r) misses Sphere(o,r) by TOPREAL9:19;
  then y <> p by A1,A2,A4,A6,XBOOLE_0:3;
  hence thesis by A4,A5,A7,A8,BROUWER:def 3;
end;
