let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) < (S-min (L~ (Cage C,n))) .. (Lower_Seq C,n)
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) < (S-min (L~ (Cage C,n))) .. (Lower_Seq C,n)
set Ema = E-max (L~ (Cage C,n));
set Sma = S-max (L~ (Cage C,n));
set Smi = S-min (L~ (Cage C,n));
set Wmi = W-min (L~ (Cage C,n));
set Rot = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
A1: Lower_Seq C,n = (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) -: (W-min (L~ (Cage C,n))) by Th26;
A2: L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = L~ (Cage C,n) by REVROT_1:33;
then A3: W-min (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by SPRECT_2:47;
E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
then (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 = E-max (L~ (Cage C,n)) by FINSEQ_6:98;
then A4: ( (S-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) < (S-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) & (S-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) <= (W-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) ) by A2, SPRECT_5:41, SPRECT_5:42;
A5: S-min (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by A2, SPRECT_2:45;
S-max (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by A2, SPRECT_2:46;
then (S-max (L~ (Cage C,n))) .. ((Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) -: (W-min (L~ (Cage C,n)))) = (S-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by A3, A4, SPRECT_5:3, XXREAL_0:2;
hence (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) < (S-min (L~ (Cage C,n))) .. (Lower_Seq C,n) by A1, A3, A4, A5, SPRECT_5:3; :: thesis: verum