theorem Th5: :: TOPREAL2:5

for P being non empty Subset of (TOP-REAL 2) holds
( P is being_simple_closed_curve iff ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) & ( for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) ) )