reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem Th50:
  for P being Subset of TOP-REAL 2 st P is being_simple_closed_curve holds
  Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) &
  Upper_Arc(P) is_an_arc_of E-max(P),W-min(P) &
  Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) &
  Lower_Arc(P) is_an_arc_of W-min(P),E-max(P) &
  Upper_Arc(P) /\ Lower_Arc(P)={W-min(P),E-max(P)}
  & Upper_Arc(P) \/ Lower_Arc(P)=P
  & First_Point(Upper_Arc(P),W-min(P),E-max(P),
  Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
  Last_Point(Lower_Arc(P),E-max(P),W-min(P),
  Vertical_Line((W-bound(P)+E-bound(P))/2))`2
proof
  let P be Subset of TOP-REAL 2;
  assume
A1: P is being_simple_closed_curve;
  then
A2: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by Def8;
  Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A1,Def9;
  hence thesis by A1,A2,Def9,JORDAN5B:14;
end;
