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))) .. (Upper_Seq C,n) = len (Upper_Seq C,n)
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: (E-max (L~ (Cage C,n))) .. (Upper_Seq C,n) = len (Upper_Seq 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)))) ;
then E-max (L~ (Cage C,n)) in rng (Upper_Seq C,n) by A1, A2, FINSEQ_5:49;
then A3: Upper_Seq C,n just_once_values E-max (L~ (Cage C,n)) by FINSEQ_4:10;
(Upper_Seq C,n) /. (len (Upper_Seq C,n)) = E-max (L~ (Cage C,n)) by JORDAN1F:7;
hence (E-max (L~ (Cage C,n))) .. (Upper_Seq C,n) = len (Upper_Seq C,n) by A3, REVROT_1:1; :: thesis: verum