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
  for f being continuous Function of Tdisk(o,r), Tdisk(o,r) holds f
  is without_fixpoints implies (BR-map(f)) | Sphere(o,r) = id
Tcircle(o,r)
proof
  let f be continuous Function of Tdisk(o,r), Tdisk(o,r) such that
A1: f is without_fixpoints;
  set D = cl_Ball(o,r);
  set C = Sphere(o,r);
  set g = BR-map(f);
  dom g = the carrier of Tdisk(o,r) & the carrier of Tdisk(o,r) = D by Th3,
FUNCT_2:def 1;
  then
A2: dom (g|C) = C by RELAT_1:62,TOPREAL9:17;
A3: the carrier of Tcircle(o,r) = C by TOPREALB:9;
A4: for x being object st x in dom (g|C) holds (g|C).x = (id Tcircle(o,r)).x
  proof
    let x be object such that
A5: x in dom (g|C);
    x in dom g by A5,RELAT_1:57;
    then reconsider y = x as Point of Tdisk(o,r);
A6: not x is_a_fixpoint_of f by A1;
    thus (g|C).x = g.x by A5,FUNCT_1:49
      .= y by A3,A5,A6,Th11
      .= (id Tcircle(o,r)).x by A3,A5,FUNCT_1:18;
  end;
  dom id Tcircle(o,r) = the carrier of Tcircle(o,r);
  hence thesis by A2,A3,A4,FUNCT_1:2;
end;
