theorem Th18: :: JORDAN20:18

for P being non empty Subset of (TOP-REAL 2)
for Q being Subset of ((TOP-REAL 2) | P)
for p1, p2, q being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} holds
( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) )