let n be Element of NAT ; :: thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-max (L~ (Cage C,n)) in rng (Upper_Seq C,n) & E-max (L~ (Cage C,n)) in L~ (Upper_Seq C,n) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( E-max (L~ (Cage C,n)) in rng (Upper_Seq C,n) & E-max (L~ (Cage C,n)) in L~ (Upper_Seq C,n) )
set x = E-max (L~ (Cage C,n));
set p = W-min (L~ (Cage C,n));
A1: Upper_Seq C,n = (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) -: (E-max (L~ (Cage C,n))) by JORDAN1E:def 1;
E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
then A2: E-max (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by FINSEQ_6:96, SPRECT_2:47;
(E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) <= (E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) ;
hence A3: E-max (L~ (Cage C,n)) in rng (Upper_Seq C,n) by A1, A2, FINSEQ_5:49; :: thesis: E-max (L~ (Cage C,n)) in L~ (Upper_Seq C,n)
len (Upper_Seq C,n) >= 2 by TOPREAL1:def 10;
then rng (Upper_Seq C,n) c= L~ (Upper_Seq C,n) by SPPOL_2:18;
hence E-max (L~ (Cage C,n)) in L~ (Upper_Seq C,n) by A3; :: thesis: verum