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:
set wi = width (Gauge C,n);
assume that
A1: ( 1 <= i & i < len (Gauge C,n) ) and
A2: (Gauge C,n) * i,(width (Gauge C,n)) in rng (Lower_Seq C,n) ; :: thesis:
consider i2 being Nat such that
A3: i2 in dom (Lower_Seq C,n) and
A4: (Lower_Seq C,n) . i2 = (Gauge C,n) * i,(width (Gauge C,n)) by A2, FINSEQ_2:11;
reconsider i2 = i2 as Element of NAT by ORDINAL1:def 13;
A5: ( 1 <= i2 & i2 <= len (Lower_Seq C,n) ) by A3, FINSEQ_3:27;
3 <= len (Upper_Seq C,n) by JORDAN1E:19;
then A6: 2 <= len (Upper_Seq C,n) by XXREAL_0:2;
set f = Rotate (Cage C,n),(E-max (L~ (Cage C,n)));
set i1 = (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n);
A7: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
E-max (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:50;
then A8: (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) /. 1 = E-max (L~ (Cage C,n)) by FINSEQ_6:98;
L~ (Cage C,n) = L~ (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) by REVROT_1:33;
then A9: ( (S-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) < (S-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) & (S-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) <= (W-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) ) by A8, SPRECT_5:41, SPRECT_5:42;
A10: ( W-min (L~ (Cage C,n)) in rng (Cage C,n) & rng (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) = rng (Cage C,n) ) by FINSEQ_6:96, SPRECT_2:47, SPRECT_2:50;
(W-min (L~ (Cage C,n))) .. (Lower_Seq C,n) = len (Lower_Seq C,n) by Th38;
then (S-min (L~ (Cage C,n))) .. (Lower_Seq C,n) <= len (Lower_Seq C,n) by Th37;
then A11: (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) < len (Lower_Seq C,n) by Th36, XXREAL_0:2;
( (E-max (L~ (Cage C,n))) .. (Lower_Seq C,n) = 1 & (E-min (L~ (Cage C,n))) .. (Lower_Seq C,n) <= (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) ) by Th33, Th35;
then A12: (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) > 1 by Th34, XXREAL_0:2;
then A13: (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) in dom (Lower_Seq C,n) by A11, FINSEQ_3:27;
( Lower_Seq C,n = (Rotate (Cage C,n),(E-max (L~ (Cage C,n)))) -: (W-min (L~ (Cage C,n))) & S-max (L~ (Cage C,n)) in rng (Cage C,n) ) by Th26, SPRECT_2:46;
then A14: S-max (L~ (Cage C,n)) in rng (Lower_Seq C,n) by A10, A9, FINSEQ_5:49, XXREAL_0:2;
then A15: (Lower_Seq C,n) /. ((S-max (L~ (Cage C,n))) .. (Lower_Seq C,n)) = S-max (L~ (Cage C,n)) by FINSEQ_5:41;
A16: (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) <> i2

assume (S-max (L~ (Cage C,n))) .. (Lower_Seq C,n) = i2 ; :: thesis:
then (Gauge C,n) * i,(width (Gauge C,n)) = S-max (L~ (Cage C,n)) by A4, A13, A15, PARTFUN1:def 8;
then ((Gauge C,n) * i,(width (Gauge C,n))) `2 = S-bound (L~ (Cage C,n)) by EUCLID:56;
then N-bound (L~ (Cage C,n)) = S-bound (L~ (Cage C,n)) by A1, A7, JORDAN1A:91;
hence contradiction by SPRECT_1:18; :: thesis:

end;

then mid (Lower_Seq C,n),((S-max (L~ (Cage C,n))) .. (Lower_Seq C,n)),i2 is being_S-Seq by A12, A11, A5, JORDAN3:39;
then reconsider h = mid (Lower_Seq C,n),((S-max (L~ (Cage C,n))) .. (Lower_Seq C,n)),i2 as one-to-one special FinSequence of (TOP-REAL 2) ;
A17: (h /. 1) `2 = ((Lower_Seq C,n) /. ((S-max (L~ (Cage C,n))) .. (Lower_Seq C,n))) `2 by A3, A13, SPRECT_2:12
.= (S-max (L~ (Cage C,n))) `2 by A14, FINSEQ_5:41
.= S-bound (L~ (Cage C,n)) by EUCLID:56 ;
len h >= 1 by A3, A13, SPRECT_2:9;
then len h > 1 by A3, A13, A16, SPRECT_2:10, XXREAL_0:1;
then A18: 1 + 1 <= len h by NAT_1:13;
A19: h is_in_the_area_of Cage C,n by A3, A13, JORDAN1E:22, SPRECT_2:26;
(h /. (len h)) `2 = ((Lower_Seq C,n) /. i2) `2 by A3, A13, SPRECT_2:13
.= ((Gauge C,n) * i,(width (Gauge C,n))) `2 by A3, A4, PARTFUN1:def 8
.= N-bound (L~ (Cage C,n)) by A1, A7, JORDAN1A:91 ;
then h is_a_v.c._for Cage C,n by A19, A17, SPRECT_2:def 3;
then L~ (Upper_Seq C,n) meets L~ h by A6, A18, Th48, SPRECT_2:33;
then consider x being set such that
A20: x in L~ (Upper_Seq C,n) and
A21: x in L~ h by XBOOLE_0:3;
L~ (mid (Lower_Seq C,n),((S-max (L~ (Cage C,n))) .. (Lower_Seq C,n)),i2) c= L~ (Lower_Seq C,n) by A12, A11, A5, JORDAN4:47;
then x in (L~ (Lower_Seq C,n)) /\ (L~ (Upper_Seq C,n)) by A20, A21, XBOOLE_0:def 4;
then A22: x in {(W-min (L~ (Cage C,n))),(E-max (L~ (Cage C,n)))} by JORDAN1E:20;
4 <= len (Gauge C,n) by JORDAN8:13;
then A23: 1 <= len (Gauge C,n) by XXREAL_0:2;

per cases by A22, TARSKI:def 2;

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

then x = (Lower_Seq C,n) /. 1 by JORDAN1F:6;
then i2 = 1 by A12, A11, A5, A21, Th45;
then (Lower_Seq C,n) /. 1 = (Gauge C,n) * i,(width (Gauge C,n)) by A3, A4, PARTFUN1:def 8;
then E-max (L~ (Cage C,n)) = (Gauge C,n) * i,(width (Gauge C,n)) by JORDAN1F:6;
then ((Gauge C,n) * i,(width (Gauge C,n))) `1 = E-bound (L~ (Cage C,n)) by EUCLID:56
.= ((Gauge C,n) * (len (Gauge C,n)),(width (Gauge C,n))) `1 by A7, A23, JORDAN1A:92 ;
hence contradiction by A1, A7, A23, GOBOARD5:4; :: thesis:

end;

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

then x = (Lower_Seq C,n) /. (len (Lower_Seq C,n)) by JORDAN1F:8;
then i2 = len (Lower_Seq C,n) by A12, A11, A5, A21, Th46;
then (Lower_Seq C,n) /. (len (Lower_Seq C,n)) = (Gauge C,n) * i,(width (Gauge C,n)) by A3, A4, PARTFUN1:def 8;
then A24: W-min (L~ (Cage C,n)) = (Gauge C,n) * i,(width (Gauge C,n)) by JORDAN1F:8;
(NW-corner (L~ (Cage C,n))) `2 >= (W-max (L~ (Cage C,n))) `2 by PSCOMP_1:87;
then (NW-corner (L~ (Cage C,n))) `2 > (W-min (L~ (Cage C,n))) `2 by SPRECT_2:61, XXREAL_0:2;
then N-bound (L~ (Cage C,n)) > ((Gauge C,n) * i,(width (Gauge C,n))) `2 by A24, EUCLID:56;
hence contradiction by A1, A7, JORDAN1A:91; :: thesis:

end;

end;