let P be Subset of (TOP-REAL 2); :: thesis: ( P is being_special_polygon implies P is being_simple_closed_curve )
given p1, p2 being Point of (TOP-REAL 2), P1, P2 being Subset of (TOP-REAL 2) such that A1: ( p1 <> p2 & p1 in P & p2 in P ) and
A2: ( P1 is_S-P_arc_joining p1,p2 & P2 is_S-P_arc_joining p1,p2 ) and
A3: ( P1 /\ P2 = {p1,p2} & P = P1 \/ P2 ) ; :: according to TOPREAL4:def 2 :: thesis: P is being_simple_closed_curve
reconsider P1 = P1, P2 = P2 as non empty Subset of (TOP-REAL 2) by A3;
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 ) by A2, Th3;
hence P is being_simple_closed_curve by A1, A3, TOPREAL2:6; :: thesis: verum