let P be Simple_closed_curve; :: thesis:
let a be Point of (TOP-REAL 2); :: thesis:
assume A1: LE a, W-min P,P ; :: thesis:

per cases by A1, JORDAN6:def 10;

suppose ( ( a in Upper_Arc P & W-min P in Lower_Arc P & not W-min P = W-min P ) or ( a in Lower_Arc P & W-min P in Lower_Arc P & not W-min P = W-min P & LE a, W-min P, Lower_Arc P, E-max P, W-min P ) ) ; :: thesis:

hence a in Lower_Arc P ; :: thesis:

end;

suppose that A2: a in Upper_Arc P and
W-min P in Upper_Arc P and
A3: LE a, W-min P, Upper_Arc P, W-min P, E-max P ; :: thesis:

Upper_Arc P is_an_arc_of W-min P, E-max P by JORDAN6:def 8;
then consider f being Function of I[01],((TOP-REAL 2) | (Upper_Arc P)), r being Real such that
A4: f is being_homeomorphism and
A5: f . 0 = W-min P and
A6: f . 1 = E-max P and
A7: 0 <= r and
A8: r <= 1 and
A9: f . r = a by A2, Th1;
thus a in Lower_Arc P :: thesis:

proof

per cases ;

suppose r = 0 ; :: thesis:

hence a in Lower_Arc P by A5, A9, JORDAN7:1; :: thesis:

end;

suppose r <> 0 ; :: thesis:

then r > 0 by A7, XXREAL_0:1;
hence a in Lower_Arc P by A3, A4, A5, A6, A8, A9, JORDAN5C:def 3; :: thesis:

end;

end;

end;

end;

end;