
theorem
  for D being Simple_closed_curve, p, q being Point of TOP-REAL 2 st p
  in D & q in D holds (TOP-REAL 2) | (D \ {p}), (TOP-REAL 2) | (D \ {q})
  are_homeomorphic
proof
  let D be Simple_closed_curve, p, q be Point of TOP-REAL 2;
  assume that
A1: p in D and
A2: q in D;
  per cases;
  suppose
    p = q;
    hence thesis;
  end;
  suppose
    p <> q;
    then reconsider
    Dp = D \ {p}, Dq = D \ {q} as non empty Subset of TOP-REAL 2 by A1,A2,
ZFMISC_1:56;
A3: (TOP-REAL 2) | Dq, I(01) are_homeomorphic by A2,Th49;
    (TOP-REAL 2) | Dp, I(01) are_homeomorphic by A1,Th49;
    hence thesis by A3,BORSUK_3:3;
  end;
end;
