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

theorem Th5:
  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 & LE q1,q2,P holds q1 in
  P & q2 in 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: LE q1,q2,P;
A3: Upper_Arc(P) \/ Lower_Arc(P)=P by A1,JORDAN6:50;
  per cases by A2,JORDAN6:def 10;
  suppose
    q1 in Upper_Arc(P) & q2 in Lower_Arc(P);
    hence thesis by A3,XBOOLE_0:def 3;
  end;
  suppose
    q1 in Upper_Arc(P) & q2 in Upper_Arc(P);
    hence thesis by A3,XBOOLE_0:def 3;
  end;
  suppose
    q1 in Lower_Arc(P) & q2 in Lower_Arc(P);
    hence thesis by A3,XBOOLE_0:def 3;
  end;
end;
