let P be Subset of (TOP-REAL 2); :: thesis:
let q1, q2, q3 be Point of (TOP-REAL 2); :: thesis:
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: LE q2,q3,P ; :: thesis:
A4: Lower_Arc P is_an_arc_of E-max P, W-min P by A1, Def9;
A5: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A1, Def9;
A6: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, Th50;

now :: thesis:

per cases by A2;

case A7: ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) ; :: thesis:

now :: thesis:

per cases by A3;

case ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) ; :: thesis:

hence LE q1,q3,P by A7; :: thesis:

end;

case A8: ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P ) ; :: thesis:

then q2 in (Upper_Arc P) /\ (Lower_Arc P) by A7, XBOOLE_0:def 4;
then q2 = E-max P by A5, A7, TARSKI:def 2;
hence LE q1,q3,P by A2, A6, A8, Th55; :: thesis:

end;

case ( q2 in Lower_Arc P & q3 in Lower_Arc P & not q3 = W-min P & LE q2,q3, Lower_Arc P, E-max P, W-min P ) ; :: thesis:

hence LE q1,q3,P by A7; :: thesis:

end;

end;

end;

hence LE q1,q3,P ; :: thesis:

end;

case A9: ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) ; :: thesis:

now :: thesis:

per cases by A3;

case ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) ; :: thesis:

hence LE q1,q3,P by A9; :: thesis:

end;

case ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P ) ; :: thesis:

then LE q1,q3, Upper_Arc P, W-min P, E-max P by A9, JORDAN5C:13;
hence LE q1,q3,P ; :: thesis:

end;

case ( q2 in Lower_Arc P & q3 in Lower_Arc P & not q3 = W-min P & LE q2,q3, Lower_Arc P, E-max P, W-min P ) ; :: thesis:

hence LE q1,q3,P by A9; :: thesis:

end;

end;

end;

hence LE q1,q3,P ; :: thesis:

end;

case A10: ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ; :: thesis:

now :: thesis:

per cases by A3;

case A11: ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) ; :: thesis:

then q2 in (Upper_Arc P) /\ (Lower_Arc P) by A10, XBOOLE_0:def 4;
then q2 = E-max P by A5, A10, TARSKI:def 2;
then LE q2,q3, Lower_Arc P, E-max P, W-min P by A4, A11, JORDAN5C:10;
then LE q1,q3, Lower_Arc P, E-max P, W-min P by A10, JORDAN5C:13;
hence LE q1,q3,P by A11; :: thesis:

end;

case A12: ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P ) ; :: thesis:

then q2 in (Upper_Arc P) /\ (Lower_Arc P) by A10, XBOOLE_0:def 4;
then q2 = E-max P by A5, A10, TARSKI:def 2;
hence LE q1,q3,P by A2, A6, A12, Th55; :: thesis:

end;

case A13: ( q2 in Lower_Arc P & q3 in Lower_Arc P & not q3 = W-min P & LE q2,q3, Lower_Arc P, E-max P, W-min P ) ; :: thesis:

then LE q1,q3, Lower_Arc P, E-max P, W-min P by A10, JORDAN5C:13;
hence LE q1,q3,P by A13; :: thesis:

end;

end;

end;

hence LE q1,q3,P ; :: thesis:

end;

end;

end;

hence LE q1,q3,P ; :: thesis: