reserve a for set;
reserve p,p1,p2,q,q1,q2 for Point of TOP-REAL 2;
reserve h1,h2 for FinSequence of TOP-REAL 2;

theorem
  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, P1,P2 being
  non empty Subset of TOP-REAL 2 st p1 <> p2 & p1 in P & p2 in P & P1
is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2}
proof
  let P be non empty Subset of TOP-REAL 2;
  hereby
    assume
A1: P is being_simple_closed_curve;
    then consider p1,p2 such that
A2: p1 <> p2 and
A3: p1 in P and
A4: p2 in P by Th5;
    consider P1,P2 being non empty Subset of TOP-REAL 2 such that
A5: P1 is_an_arc_of p1,p2 and
A6: P2 is_an_arc_of p1,p2 and
A7: P = P1 \/ P2 and
A8: P1 /\ P2 = {p1,p2} by A1,A2,A3,A4,Th5;
    take p1,p2,P1,P2;
    thus p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2
is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} by A2,A3,A4,A5,A6,A7,A8;
  end;
  thus thesis by Lm34;
end;
