reserve P,P1,P2,R for Subset of TOP-REAL 2,
  p,p1,p2,p3,p11,p22,q,q1,q2,q3,q4 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r for Real,
  u for Point of Euclid 2,
  n,m,i,j,k for Nat,
  x,y for set;

theorem
  P is being_special_polygon implies P is being_simple_closed_curve
proof
  given p1,p2,P1,P2 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} and
A4: P = P1 \/ P2;
  reconsider P1, 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,Th2;
  hence thesis by A1,A3,A4,TOPREAL2:6;
end;
