let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: 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; :: thesis: 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; :: thesis: verum

let n be Nat; :: thesis: 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; :: thesis: verum