let n be Nat; :: thesis:
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
let i be 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_2:10;
reconsider i2 = i2 as Nat ;
A5: ( 1 <= i2 & i2 <= len (Lower_Seq (C,n)) ) by A3, FINSEQ_3:25;
3 <= len (Upper_Seq (C,n)) by JORDAN1E:15;
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_2:46;
then A8: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92;
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_5:40, SPRECT_5:41;
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_6:90, SPRECT_2:43, SPRECT_2:46;
(W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) by Th30;
then (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by Th29;
then A11: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by Th28, 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 Th25, Th27;
then A12: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) > 1 by Th26, XXREAL_0:2;
then A13: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
( 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 Th18, SPRECT_2:42;
then A14: S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by A10, A9, FINSEQ_5:46, 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_5:38;
A16: ( (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) in NAT & i2 in NAT ) by ORDINAL1:def 12;
A17: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <> i2

proof

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 6;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
then N-bound (L~ (Cage (C,n))) = S-bound (L~ (Cage (C,n))) by A1, A7, JORDAN1A:70;
hence contradiction by SPRECT_1:16; :: 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, JORDAN3:6, A16;
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) ;
A18: (h /. 1) `2 = ((Lower_Seq (C,n)) /. ((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)))) `2 by A3, A13, SPRECT_2:8
.= (S-max (L~ (Cage (C,n)))) `2 by A14, FINSEQ_5:38
.= S-bound (L~ (Cage (C,n))) by EUCLID:52 ;
len h >= 1 by A3, A13, SPRECT_2:5;
then len h > 1 by A3, A13, A17, SPRECT_2:6, XXREAL_0:1;
then A19: 1 + 1 <= len h by NAT_1:13;
A20: h is_in_the_area_of Cage (C,n) by A3, A13, JORDAN1E:18, SPRECT_2:22;
(h /. (len h)) `2 = ((Lower_Seq (C,n)) /. i2) `2 by A3, A13, SPRECT_2:9
.= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A3, A4, PARTFUN1:def 6
.= N-bound (L~ (Cage (C,n))) by A1, A7, JORDAN1A:70 ;
then h is_a_v.c._for Cage (C,n) by A20, A18, SPRECT_2:def 3;
then L~ (Upper_Seq (C,n)) meets L~ h by A6, A19, Th40, SPRECT_2:29;
then consider x being object such that
A21: x in L~ (Upper_Seq (C,n)) and
A22: 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, JORDAN4:35;
then x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A21, A22, XBOOLE_0:def 4;
then A23: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
4 <= len (Gauge (C,n)) by JORDAN8:10;
then A24: 1 <= len (Gauge (C,n)) by XXREAL_0:2;

per cases by A23, 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, A22, Th37;
then (Lower_Seq (C,n)) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A3, A4, PARTFUN1:def 6;
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:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `1 by A7, A24, JORDAN1A:71 ;
hence contradiction by A1, A7, A24, GOBOARD5:3; :: 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, A22, Th38;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A3, A4, PARTFUN1:def 6;
then A25: 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_1:30;
then (NW-corner (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2 by SPRECT_2:57, XXREAL_0:2;
then N-bound (L~ (Cage (C,n))) > ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A25, EUCLID:52;
hence contradiction by A1, A7, JORDAN1A:70; :: thesis:

end;