let P be Simple_closed_curve; :: thesis: for a being Point of (TOP-REAL 2) st LE E-max P,a,P holds
a in Lower_Arc P
let a be Point of (TOP-REAL 2); :: thesis: ( LE E-max P,a,P implies a in Lower_Arc P )
assume A1: LE E-max P,a,P ; :: thesis: a in Lower_Arc P

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 A1, JORDAN6:def 10;

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: a in Lower_Arc P

hence a in Lower_Arc P ; :: thesis: verum

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: a in Lower_Arc P

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 & 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 A2, Th1;
thus a in Lower_Arc P :: thesis: verum

proof

per cases ( r = 1 or r <> 1 ) ;

suppose r = 1 ; :: thesis: a in Lower_Arc P

hence a in Lower_Arc P by A5, A8, JORDAN7:1; :: thesis: verum

end;

suppose r <> 1 ; :: thesis: a in Lower_Arc P

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

end;

end;

end;

end;

end;