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

theorem Th7:
  for P being compact non empty Subset of TOP-REAL 2, q1 being
  Point of TOP-REAL 2 st q1 in P & P is being_simple_closed_curve holds q1 in
  Segment(q1,W-min P,P)
proof
  let P be compact non empty Subset of TOP-REAL 2, q1 be Point of TOP-REAL 2
  such that
A1: q1 in P;
  assume P is being_simple_closed_curve;
  then LE q1,q1,P by A1,JORDAN6:56;
  then q1 in {p1: LE q1,p1,P or q1 in P & p1=W-min(P)};
  hence thesis by Def1;
end;
