let n be Element of NAT ; :: thesis:
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
let i be Element of NAT ; :: thesis:
assume that
A1: ( 1 < i & i <= len (Gauge C,n) ) and
A2: (Gauge C,n) * i,1 in rng (Upper_Seq C,n) ; :: thesis:
consider i2 being Nat such that
A3: i2 in dom (Upper_Seq C,n) and
A4: (Upper_Seq C,n) . i2 = (Gauge C,n) * i,1 by A2, FINSEQ_2:11;
reconsider i2 = i2 as Element of NAT by ORDINAL1:def 13;
A5: ( 1 <= i2 & i2 <= len (Upper_Seq C,n) ) by A3, FINSEQ_3:27;
set f = Rotate (Cage C,n),(W-min (L~ (Cage C,n)));
set i1 = (N-min (L~ (Cage C,n))) .. (Upper_Seq C,n);
A6: ( E-max (L~ (Cage C,n)) in rng (Cage C,n) & rng (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) = rng (Cage C,n) ) by FINSEQ_6:96, SPRECT_2:47, SPRECT_2:50;
W-min (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:47;
then A7: (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 1 = W-min (L~ (Cage C,n)) by FINSEQ_6:98;
L~ (Cage C,n) = L~ (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by REVROT_1:33;
then A8: ( (N-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) < (N-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) & (N-max (L~ (Cage C,n))) .. (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)))) ) by A7, SPRECT_5:25, SPRECT_5:26;
(E-max (L~ (Cage C,n))) .. (Upper_Seq C,n) = len (Upper_Seq C,n) by Th32;
then (N-max (L~ (Cage C,n))) .. (Upper_Seq C,n) <= len (Upper_Seq C,n) by Th31;
then A9: (N-min (L~ (Cage C,n))) .. (Upper_Seq C,n) < len (Upper_Seq C,n) by Th30, XXREAL_0:2;
3 <= len (Lower_Seq C,n) by JORDAN1E:19;
then A10: 2 <= len (Lower_Seq C,n) by XXREAL_0:2;
A11: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
4 <= len (Gauge C,n) by JORDAN8:13;
then A12: 1 <= len (Gauge C,n) by XXREAL_0:2;
( (W-min (L~ (Cage C,n))) .. (Upper_Seq C,n) = 1 & (W-max (L~ (Cage C,n))) .. (Upper_Seq C,n) <= (N-min (L~ (Cage C,n))) .. (Upper_Seq C,n) ) by Th27, Th29;
then A13: (N-min (L~ (Cage C,n))) .. (Upper_Seq C,n) > 1 by Th28, XXREAL_0:2;
then A14: (N-min (L~ (Cage C,n))) .. (Upper_Seq C,n) in dom (Upper_Seq C,n) by A9, FINSEQ_3:27;
( Upper_Seq C,n = (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) -: (E-max (L~ (Cage C,n))) & N-min (L~ (Cage C,n)) in rng (Cage C,n) ) by JORDAN1E:def 1, SPRECT_2:43;
then A15: N-min (L~ (Cage C,n)) in rng (Upper_Seq C,n) by A6, A8, FINSEQ_5:49, XXREAL_0:2;
then A16: (Upper_Seq C,n) /. ((N-min (L~ (Cage C,n))) .. (Upper_Seq C,n)) = N-min (L~ (Cage C,n)) by FINSEQ_5:41;
A17: (N-min (L~ (Cage C,n))) .. (Upper_Seq C,n) <> i2

assume (N-min (L~ (Cage C,n))) .. (Upper_Seq C,n) = i2 ; :: thesis:
then (Gauge C,n) * i,1 = N-min (L~ (Cage C,n)) by A4, A14, A16, PARTFUN1:def 8;
then ((Gauge C,n) * i,1) `2 = N-bound (L~ (Cage C,n)) by EUCLID:56;
then S-bound (L~ (Cage C,n)) = N-bound (L~ (Cage C,n)) by A1, JORDAN1A:93;
hence contradiction by SPRECT_1:18; :: thesis:

end;

then mid (Upper_Seq C,n),((N-min (L~ (Cage C,n))) .. (Upper_Seq C,n)),i2 is being_S-Seq by A13, A9, A5, JORDAN3:39;
then reconsider h1 = mid (Upper_Seq C,n),((N-min (L~ (Cage C,n))) .. (Upper_Seq C,n)),i2 as one-to-one special FinSequence of (TOP-REAL 2) ;
set h = Rev h1;
A18: len h1 = len (Rev h1) by FINSEQ_5:def 3;
then A19: not h1 is empty by A3, A14, SPRECT_2:9;
then A20: ((Rev h1) /. (len (Rev h1))) `2 = (h1 /. 1) `2 by A18, FINSEQ_5:68
.= ((Upper_Seq C,n) /. ((N-min (L~ (Cage C,n))) .. (Upper_Seq C,n))) `2 by A3, A14, SPRECT_2:12
.= (N-min (L~ (Cage C,n))) `2 by A15, FINSEQ_5:41
.= N-bound (L~ (Cage C,n)) by EUCLID:56 ;
h1 is_in_the_area_of Cage C,n by A3, A14, JORDAN1E:21, SPRECT_2:26;
then A21: Rev h1 is_in_the_area_of Cage C,n by SPRECT_3:68;
((Rev h1) /. 1) `2 = (h1 /. (len h1)) `2 by A19, FINSEQ_5:68
.= ((Upper_Seq C,n) /. i2) `2 by A3, A14, SPRECT_2:13
.= ((Gauge C,n) * i,1) `2 by A3, A4, PARTFUN1:def 8
.= S-bound (L~ (Cage C,n)) by A1, JORDAN1A:93 ;
then A22: ( Rev (Lower_Seq C,n) is special & Rev h1 is_a_v.c._for Cage C,n ) by A21, A20, SPPOL_2:42, SPRECT_2:def 3;
len (Rev h1) >= 1 by A3, A14, A18, SPRECT_2:9;
then len (Rev h1) > 1 by A3, A14, A17, A18, SPRECT_2:10, XXREAL_0:1;
then A23: 1 + 1 <= len (Rev h1) by NAT_1:13;
( len (Lower_Seq C,n) = len (Rev (Lower_Seq C,n)) & Rev h1 is special ) by FINSEQ_5:def 3, SPPOL_2:42;
then ( L~ (Rev (Lower_Seq C,n)) = L~ (Lower_Seq C,n) & L~ (Rev (Lower_Seq C,n)) meets L~ (Rev h1) ) by A10, A23, A22, Th49, SPPOL_2:22, SPRECT_2:33;
then consider x being set such that
A24: x in L~ (Lower_Seq C,n) and
A25: x in L~ (Rev h1) by XBOOLE_0:3;
A26: L~ (Rev h1) = L~ h1 by SPPOL_2:22;
L~ (mid (Upper_Seq C,n),((N-min (L~ (Cage C,n))) .. (Upper_Seq C,n)),i2) c= L~ (Upper_Seq C,n) by A13, A9, A5, JORDAN4:47;
then x in (L~ (Upper_Seq C,n)) /\ (L~ (Lower_Seq C,n)) by A24, A25, A26, XBOOLE_0:def 4;
then A27: x in {(W-min (L~ (Cage C,n))),(E-max (L~ (Cage C,n)))} by JORDAN1E:20;

per cases by A27, TARSKI:def 2;

suppose x = W-min (L~ (Cage C,n)) ; :: thesis:

then x = (Upper_Seq C,n) /. 1 by JORDAN1F:5;
then i2 = 1 by A13, A9, A5, A25, A26, Th45;
then (Upper_Seq C,n) /. 1 = (Gauge C,n) * i,1 by A3, A4, PARTFUN1:def 8;
then W-min (L~ (Cage C,n)) = (Gauge C,n) * i,1 by JORDAN1F:5;
then ((Gauge C,n) * i,1) `1 = W-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * 1,1) `1 by A12, JORDAN1A:94 ;
hence contradiction by A1, A12, A11, GOBOARD5:4; :: thesis:

end;

suppose x = E-max (L~ (Cage C,n)) ; :: thesis:

then x = (Upper_Seq C,n) /. (len (Upper_Seq C,n)) by JORDAN1F:7;
then i2 = len (Upper_Seq C,n) by A13, A9, A5, A25, A26, Th46;
then (Upper_Seq C,n) /. (len (Upper_Seq C,n)) = (Gauge C,n) * i,1 by A3, A4, PARTFUN1:def 8;
then A28: E-max (L~ (Cage C,n)) = (Gauge C,n) * i,1 by JORDAN1F:7;
(SE-corner (L~ (Cage C,n))) `2 <= (E-min (L~ (Cage C,n))) `2 by PSCOMP_1:107;
then (SE-corner (L~ (Cage C,n))) `2 < (E-max (L~ (Cage C,n))) `2 by SPRECT_2:57, XXREAL_0:2;
then S-bound (L~ (Cage C,n)) < ((Gauge C,n) * i,1) `2 by A28, EUCLID:56;
hence contradiction by A1, JORDAN1A:93; :: thesis:

end;

end;