let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for n being Element of NAT st n > 0 holds
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n)))
let n be Element of NAT ; :: thesis: ( n > 0 implies L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) )
A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:47;
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;
A3: 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 A2, FINSEQ_5:47;
then A4: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:98;
(Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A3, A2, FINSEQ_5:45
.= E-max (L~ (Cage (C,n))) by A2, FINSEQ_5:48 ;
then A5: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A4, TOPREAL1:31;
assume n > 0 ; :: thesis: L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n)))
then A6: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by Th62;
A7: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def 2
.= E-max (L~ (Cage (C,n))) by FINSEQ_5:56 ;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def 2;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A2, FINSEQ_5:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def 1
.= W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:98 ;
then A8: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by A7, TOPREAL1:31;
( (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) = {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} & (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) = L~ (Cage (C,n)) ) by JORDAN1E:17, JORDAN1E:20;
hence L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A5, A8, A6, JORDAN6:def 8; :: thesis: verum