let n be Element of NAT ; :: thesis:
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
set x = N-min (L~ (Cage C,n));
set p = W-min (L~ (Cage C,n));
set f = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
A1: rng (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = rng (Cage C,n) by FINSEQ_6:96, SPRECT_2:50;
A2: N-min (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:43;
A3: L~ (Cage C,n) = L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by REVROT_1:33;
W-min (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:47;
then A4: W-min (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by FINSEQ_6:96, SPRECT_2:50;
A5: W-min (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:47;
Lower_Seq C,n = (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) -: (W-min (L~ (Cage C,n))) by JORDAN1G:26;
then A6: (Lower_Seq C,n) /. 1 = (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 by A4, FINSEQ_5:47;
then A7: (W-max (L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) <= (N-min (L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by A3, JORDAN1F:6, SPRECT_5:44;
(W-min (L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) < (W-max (L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by A6, A3, JORDAN1F:6, SPRECT_5:43;
then N-min (L~ (Cage C,n)) in rng ((Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) :- (W-min (L~ (Cage C,n)))) by A1, A2, A5, A3, A7, FINSEQ_6:67, XXREAL_0:2;
hence A8: N-min (L~ (Cage C,n)) in rng (Upper_Seq C,n) by Th4; :: thesis:
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 N-min (L~ (Cage C,n)) in L~ (Upper_Seq C,n) by A8; :: thesis: