
theorem Th49:
  for D being Simple_closed_curve, p being Point of TOP-REAL 2 st
  p in D holds (TOP-REAL 2) | (D \ {p}), I(01) are_homeomorphic
proof
  let D be Simple_closed_curve, p be Point of TOP-REAL 2;
  consider q being Point of TOP-REAL 2 such that
A1: q in D and
A2: p <> q by SUBSET_1:51;
  not q in {p} by A2,TARSKI:def 1;
  then reconsider R2p = D \ {p} as non empty Subset of TOP-REAL 2 by A1,
XBOOLE_0:def 5;
  assume p in D;
  then consider P1, P2 being non empty Subset of TOP-REAL 2 such that
A3: P1 is_an_arc_of p,q and
A4: P2 is_an_arc_of p,q and
A5: D = P1 \/ P2 and
A6: P1 /\ P2 = {p,q} by A1,A2,TOPREAL2:5;
A7: P2 is_an_arc_of q, p by A4,JORDAN5B:14;
  D \ {p} = (P1 \ {p}) \/ (P2 \ {p}) by A5,XBOOLE_1:42;
  then (TOP-REAL 2) | R2p, I(01) are_homeomorphic by A2,A3,A6,A7,Th48;
  hence thesis;
end;
