let n be Element of NAT ; :: thesis:
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
set Ca = Cage (C,n);
set US = Upper_Seq (C,n);
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Nmin = N-min (L~ (Cage (C,n)));
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:50;
then A1: 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 (Upper_Seq (C,n)) >= 3 by JORDAN1E:19;
then len (Upper_Seq (C,n)) >= 2 by XXREAL_0:2;
then 2 in Seg (len (Upper_Seq (C,n))) by FINSEQ_1:3;
then A2: 2 in Seg ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by JORDAN1E:12;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:56;
then A3: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:46;
(Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:34;
then (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by SPRECT_2:80;
then A4: ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;
( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:56, SPRECT_1:15;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:71;
then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:61;
then A5: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:76;
A6: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:47;
then A7: 1 <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by FINSEQ_4:31;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A6, FINSEQ_5:57
.= (Cage (C,n)) /. 1 by FINSEQ_6:def 1
.= N-min (L~ (Cage (C,n))) by JORDAN9:34 ;
then A8: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; :: thesis:
hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A8, A3, TARSKI:def 2; :: thesis:

end;

then A9: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:27;
card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:80;
then card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:104;
then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A5, A9, XBOOLE_1:1;
then A10: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:40;
then A11: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 1 by XXREAL_0:2;
A12: (Upper_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def 1
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A1, FINSEQ_5:47
.= (Cage (C,n)) /. ((1 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A6, A11, REVROT_1:9
.= (Cage (C,n)) /. (0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_1:234 ;
(Upper_Seq (C,n)) /. 2 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. 2 by JORDAN1E:def 1
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by A1, A2, FINSEQ_5:46
.= (Cage (C,n)) /. ((2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A6, A10, REVROT_1:9
.= (Cage (C,n)) /. ((2 - 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_0:def 2 ;
hence ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n))) by A7, A4, A12, JORDAN1E:26, JORDAN1F:5; :: thesis: