let n be Element of NAT ; :: thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-min (L~ (Cage C,n)) in rng (Lower_Seq C,n) & E-min (L~ (Cage C,n)) in L~ (Lower_Seq C,n) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( E-min (L~ (Cage C,n)) in rng (Lower_Seq C,n) & E-min (L~ (Cage C,n)) in L~ (Lower_Seq C,n) )
set x = E-min (L~ (Cage C,n));
set p = E-max (L~ (Cage C,n));
set f = Rotate (Cage C,n),(W-min (L~ (Cage C,n)));
A1: rng (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) = rng (Cage C,n) by FINSEQ_6:96, SPRECT_2:47;
A2: E-min (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:49;
A3: L~ (Cage C,n) = L~ (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by REVROT_1:33;
E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
then A4: 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;
A5: E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
Upper_Seq C,n = (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) -: (E-max (L~ (Cage C,n))) by JORDAN1E:def 1;
then (Upper_Seq C,n) /. 1 = (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 1 by A4, FINSEQ_5:47;
then (E-max (L~ (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) < (E-min (L~ (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A3, JORDAN1F:5, SPRECT_5:27;
then E-min (L~ (Cage C,n)) in rng ((Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) :- (E-max (L~ (Cage C,n)))) by A1, A2, A5, A3, FINSEQ_6:67;
hence A6: E-min (L~ (Cage C,n)) in rng (Lower_Seq C,n) by JORDAN1E:def 2; :: thesis: E-min (L~ (Cage C,n)) in L~ (Lower_Seq C,n)
len (Lower_Seq C,n) >= 2 by TOPREAL1:def 10;
then rng (Lower_Seq C,n) c= L~ (Lower_Seq C,n) by SPPOL_2:18;
hence E-min (L~ (Cage C,n)) in L~ (Lower_Seq C,n) by A6; :: thesis: verum