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;
len (Lower_Seq (C,n)) = ((len (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)))))))) + 1 by Th13;
then ((len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - (len (Lower_Seq (C,n)))) + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A2, FINSEQ_4:31;
then (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - (len (Lower_Seq (C,n))) <= (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) - 1 by XREAL_1:21;
hence not Lower_Seq (C,n) is empty by XREAL_1:12; :: thesis: verum