reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem
  P is special_polygonal implies for p1,p2 st p1 <> p2 & p1 in P & p2 in
  P ex P1,P2 being Subset of TOP-REAL 2 st P1 is_S-P_arc_joining p1,p2 & P2
  is_S-P_arc_joining p1,p2 & P1 /\ P2 = {p1,p2} & P = P1 \/ P2
proof
  assume
A1: P is special_polygonal;
  let p1,p2;
  assume that
A2: p1 <> p2 and
A3: p1 in P and
A4: p2 in P;
  p1,p2 split P by A1,A2,A3,A4,Th58;
  then consider f1,f2 being S-Sequence_in_R2 such that
A5: p1 = f1/.1 and
A6: p1 = f2/.1 and
A7: p2 = f1/.len f1 and
A8: p2 = f2/.len f2 and
A9: L~f1 /\ L~f2 = {p1,p2} and
A10: P = L~f1 \/ L~f2;
  take P1 = L~f1, P2 = L~f2;
  thus P1 is_S-P_arc_joining p1,p2 by A5,A7;
  thus P2 is_S-P_arc_joining p1,p2 by A6,A8;
  thus thesis by A9,A10;
end;
