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 Th3:
  P is_S-P_arc_joining p,q implies p in P & q in P
proof
  assume P is_S-P_arc_joining p,q;
  then P is_an_arc_of p,q by Th2;
  hence thesis by TOPREAL1:1;
end;
