let P be Subset of (); :: thesis: for q1, q2, q3 being Point of () st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P holds
LE q1,q3,P

let q1, q2, q3 be Point of (); :: thesis: ( P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P implies LE q1,q3,P )
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: LE q2,q3,P ; :: thesis: LE q1,q3,P
A4: Lower_Arc P is_an_arc_of E-max P, W-min P by ;
A5: (Upper_Arc P) /\ () = {(),()} by ;
A6: Upper_Arc P is_an_arc_of W-min P, E-max P by ;
now :: thesis: ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q3,P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P & LE q1,q3,P ) or ( 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 & LE q1,q3,P ) )
per cases ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( 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 ) ) by A2;
case A7: ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) ; :: thesis: LE q1,q3,P
now :: thesis: ( ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P & LE q1,q3,P ) or ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P & LE q1,q3,P ) or ( 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 & LE q1,q3,P ) )
per cases ( ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) or ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P ) or ( 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 ) ) by A3;
case ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) ; :: thesis: LE q1,q3,P
hence LE q1,q3,P by A7; :: thesis: verum
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: LE q1,q3,P
then q2 in () /\ () by ;
then q2 = E-max P by ;
hence LE q1,q3,P by A2, A6, A8, Th55; :: thesis: verum
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: LE q1,q3,P
hence LE q1,q3,P by A7; :: thesis: verum
end;
end;
end;
hence LE q1,q3,P ; :: thesis: verum
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: LE q1,q3,P
now :: thesis: ( ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P & LE q1,q3,P ) or ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P & LE q1,q3,P ) or ( 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 & LE q1,q3,P ) )
per cases ( ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) or ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P ) or ( 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 ) ) by A3;
case ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) ; :: thesis: LE q1,q3,P
hence LE q1,q3,P by A9; :: thesis: verum
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: LE q1,q3,P
then LE q1,q3, Upper_Arc P, W-min P, E-max P by ;
hence LE q1,q3,P ; :: thesis: verum
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: LE q1,q3,P
hence LE q1,q3,P by A9; :: thesis: verum
end;
end;
end;
hence LE q1,q3,P ; :: thesis: verum
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: LE q1,q3,P
now :: thesis: ( ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P & LE q1,q3,P ) or ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P & LE q1,q3,P ) or ( 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 & LE q1,q3,P ) )
per cases ( ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) or ( q2 in Upper_Arc P & q3 in Upper_Arc P & LE q2,q3, Upper_Arc P, W-min P, E-max P ) or ( 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 ) ) by A3;
case A11: ( q2 in Upper_Arc P & q3 in Lower_Arc P & not q3 = W-min P ) ; :: thesis: LE q1,q3,P
then q2 in () /\ () by ;
then q2 = E-max P by ;
then LE q2,q3, Lower_Arc P, E-max P, W-min P by ;
then LE q1,q3, Lower_Arc P, E-max P, W-min P by ;
hence LE q1,q3,P by A11; :: thesis: verum
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: LE q1,q3,P
then q2 in () /\ () by ;
then q2 = E-max P by ;
hence LE q1,q3,P by A2, A6, A12, Th55; :: thesis: verum
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: LE q1,q3,P
then LE q1,q3, Lower_Arc P, E-max P, W-min P by ;
hence LE q1,q3,P by A13; :: thesis: verum
end;
end;
end;
hence LE q1,q3,P ; :: thesis: verum
end;
end;
end;
hence LE q1,q3,P ; :: thesis: verum