let D be Simple_closed_curve; for p being Point of (TOP-REAL 2) st p in D holds
(TOP-REAL 2) | (D \ {p}), I(01) are_homeomorphic
let p be Point of (TOP-REAL 2); ( p in D implies (TOP-REAL 2) | (D \ {p}), I(01) are_homeomorphic )
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
; (TOP-REAL 2) | (D \ {p}), I(01) are_homeomorphic
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
(TOP-REAL 2) | (D \ {p}), I(01) are_homeomorphic
; verum