reserve p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem
  for P being compact non empty Subset of TOP-REAL 2, q1,q2 being Point
  of TOP-REAL 2 st P is being_simple_closed_curve & q1<>W-min(P) & q2<>W-min(P)
  holds not W-min(P) in Segment(q1,q2,P)
proof
  let P be compact non empty Subset of TOP-REAL 2, q1,q2 be Point of TOP-REAL
  2;
  assume that
A1: P is being_simple_closed_curve and
A2: q1<>W-min(P) and
A3: q2<>W-min(P);
A4: Segment(q1,q2,P)={p: LE q1,p,P & LE p,q2,P} by A3,Def1;
  now
A5: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A1,JORDAN6:def 8;
    assume W-min(P) in Segment(q1,q2,P);
    then consider p such that
A6: p=W-min(P) and
A7: LE q1,p,P and
    LE p,q2,P by A4;
    LE q1,p,Upper_Arc(P),W-min(P),E-max(P) by A6,A7,JORDAN6:def 10;
    hence contradiction by A2,A6,A5,JORDAN6:54;
  end;
  hence thesis;
end;
