let P be Simple_closed_curve; :: thesis: for a being Point of () st LE E-max P,a,P holds
a in Lower_Arc P

let a be Point of (); :: thesis: ( LE E-max P,a,P implies a in Lower_Arc P )
assume A1: LE E-max P,a,P ; :: thesis:
per cases ( ( E-max P in Upper_Arc P & a in Lower_Arc P & not a = W-min P ) or ( E-max P in Lower_Arc P & a in Lower_Arc P & not a = W-min P & LE E-max P,a, Lower_Arc P, E-max P, W-min P ) or ( E-max P in Upper_Arc P & a in Upper_Arc P & LE E-max P,a, Upper_Arc P, W-min P, E-max P ) ) by ;
suppose ( ( E-max P in Upper_Arc P & a in Lower_Arc P & not a = W-min P ) or ( E-max P in Lower_Arc P & a in Lower_Arc P & not a = W-min P & LE E-max P,a, Lower_Arc P, E-max P, W-min P ) ) ; :: thesis:
end;
suppose that E-max P in Upper_Arc P and
A2: a in Upper_Arc P and
A3: LE E-max P,a, 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],(() | ()), r being Real such that
A4: ( f is being_homeomorphism & f . 0 = W-min P ) and
A5: f . 1 = E-max P and
A6: 0 <= r and
A7: r <= 1 and
A8: f . r = a by ;
thus a in Lower_Arc P :: thesis: verum
end;
end;