let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Upper_Seq C,n) /. 2) `1 = W-bound (L~ (Cage C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ((Upper_Seq C,n) /. 2) `1 = W-bound (L~ (Cage C,n))
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: ( not x in {(N-min (L~ (Cage C,n))),(W-min (L~ (Cage C,n)))} or x in rng ((Cage C,n) :- (W-min (L~ (Cage C,n)))) )
assume x in {(N-min (L~ (Cage C,n))),(W-min (L~ (Cage C,n)))} ; :: thesis: x in rng ((Cage C,n) :- (W-min (L~ (Cage C,n))))
hence x in rng ((Cage C,n) :- (W-min (L~ (Cage C,n)))) by A8, A3, TARSKI:def 2; :: thesis: verum

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: verum