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 Th65:
  for n being non zero Element of NAT, o, p being Point of TOP-REAL n,
  r being positive Real st p in Ball(o,r) holds
  DiskProj(o,r,p)|Sphere(o,r) = id Sphere(o,r)
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);
A2: the carrier of Tdisk(o,r) = cl_Ball(o,r) by BROUWER:3;
A3: the carrier of (TOP-REAL n)|(cl_Ball(o,r)\{p}) = cl_Ball(o,r)\{p}
  by PRE_TOPC:8;
A4: dom DiskProj(o,r,p) = the carrier of (TOP-REAL n)|(cl_Ball(o,r)\{p})
  by FUNCT_2:def 1;
A5: Sphere(o,r) misses Ball(o,r) by TOPREAL9:19;
A6: Sphere(o,r) c= cl_Ball(o,r) by TOPREAL9:17;
A7: Ball(o,r) c= cl_Ball(o,r) by TOPREAL9:16;
A8: Sphere(o,r) c= cl_Ball(o,r)\{p}
  proof
    let a be object;
    assume
A9: a in Sphere(o,r);
    then a <> p by A1,A5,XBOOLE_0:3;
    hence thesis by A6,A9,ZFMISC_1:56;
  end;
  then
A10: dom(DiskProj(o,r,p)|Sphere(o,r)) = Sphere(o,r) by A3,A4,RELAT_1:62;
A11: dom id Sphere(o,r) = Sphere(o,r);
  now
    let x be object;
    assume
A12: x in dom(DiskProj(o,r,p)|Sphere(o,r));
    then x in dom DiskProj(o,r,p) by RELAT_1:57;
    then consider y being Point of TOP-REAL n such that
A13: x = y and
A14: (DiskProj(o,r,p)).x = HC(p,y,o,r) by A1,A2,A7,Def7;
    y in halfline(p,y) by TOPREAL9:28;
    then
A15: x in halfline(p,y) /\ Sphere(o,r) by A12,A13,XBOOLE_0:def 4;
A16: x <> p by A1,A5,A12,XBOOLE_0:3;
    thus (DiskProj(o,r,p)|Sphere(o,r)).x = (DiskProj(o,r,p)).x
    by A12,FUNCT_1:47
      .= x by A1,A2,A6,A7,A10,A12,A13,A14,A15,A16,BROUWER:def 3
      .= (id Sphere(o,r)).x by A12,FUNCT_1:18;
  end;
  hence thesis by A3,A4,A8,A11,FUNCT_1:2,RELAT_1:62;
end;
