let C be compact non horizontal non vertical Subset of (TOP-REAL 2); for n being Nat holds 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
let n be Nat; 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
E-max (L~ (Cage (C,n))) in rng (Cage (C,n))
by SPRECT_2:46;
then
E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))
by FINSEQ_6:90, SPRECT_2:43;
hence
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 FINSEQ_5:50; verum