let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Nat st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for i being Nat st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n))
let i be Nat; :: thesis: ( 1 < i & i <= len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) )
assume that
A1: ( 1 < i & i <= len (Gauge (C,n)) ) and
A2: (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) ; :: thesis: contradiction
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_2:10;
reconsider i2 = i2 as Nat ;
A5: ( 1 <= i2 & i2 <= len (Upper_Seq (C,n)) ) by A3, FINSEQ_3:25;
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_6:90, SPRECT_2:43, SPRECT_2:46;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A7: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
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_5:24, SPRECT_5:25;
(E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by Th24;
then (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by Th23;
then A9: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < len (Upper_Seq (C,n)) by Th22, XXREAL_0:2;
3 <= len (Lower_Seq (C,n)) by JORDAN1E:15;
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 JORDAN8:10;
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 Th19, Th21;
then A13: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) > 1 by Th20, XXREAL_0:2;
then A14: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25;
( 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_2:39;
then A15: N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A6, A8, FINSEQ_5:46, 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_5:38;
A17: ( (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) in NAT & i2 in NAT ) by ORDINAL1:def 12;
A18: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <> i2

proof

assume (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = i2 ; :: thesis: contradiction
then (Gauge (C,n)) * (i,1) = N-min (L~ (Cage (C,n))) by A4, A14, A16, PARTFUN1:def 6;
then ((Gauge (C,n)) * (i,1)) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
then S-bound (L~ (Cage (C,n))) = N-bound (L~ (Cage (C,n))) by A1, JORDAN1A:72;
hence contradiction by SPRECT_1:16; :: thesis: verum

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, JORDAN3:6, A17;
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;
A19: len h1 = len (Rev h1) by FINSEQ_5:def 3;
then A20: not h1 is empty by A3, A14, SPRECT_2:5;
then A21: ((Rev h1) /. (len (Rev h1))) `2 = (h1 /. 1) `2 by A19, FINSEQ_5:65
.= ((Upper_Seq (C,n)) /. ((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)))) `2 by A3, A14, SPRECT_2:8
.= (N-min (L~ (Cage (C,n)))) `2 by A15, FINSEQ_5:38
.= N-bound (L~ (Cage (C,n))) by EUCLID:52 ;
h1 is_in_the_area_of Cage (C,n) by A3, A14, JORDAN1E:17, SPRECT_2:22;
then A22: Rev h1 is_in_the_area_of Cage (C,n) by SPRECT_3:51;
((Rev h1) /. 1) `2 = (h1 /. (len h1)) `2 by A20, FINSEQ_5:65
.= ((Upper_Seq (C,n)) /. i2) `2 by A3, A14, SPRECT_2:9
.= ((Gauge (C,n)) * (i,1)) `2 by A3, A4, PARTFUN1:def 6
.= S-bound (L~ (Cage (C,n))) by A1, JORDAN1A:72 ;
then A23: ( Rev (Lower_Seq (C,n)) is special & Rev h1 is_a_v.c._for Cage (C,n) ) by A22, A21, SPRECT_2:def 3;
len (Rev h1) >= 1 by A3, A14, A19, SPRECT_2:5;
then len (Rev h1) > 1 by A3, A14, A18, A19, SPRECT_2:6, XXREAL_0:1;
then A24: 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_2:40;
then ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & L~ (Rev (Lower_Seq (C,n))) meets L~ (Rev h1) ) by A10, A24, A23, Th41, SPPOL_2:22, SPRECT_2:29;
then consider x being object such that
A25: x in L~ (Lower_Seq (C,n)) and
A26: x in L~ (Rev h1) by XBOOLE_0:3;
A27: 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, JORDAN4:35;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A25, A26, A27, XBOOLE_0:def 4;
then A28: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;

per cases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A28, TARSKI:def 2;

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

then x = (Upper_Seq (C,n)) /. 1 by JORDAN1F:5;
then i2 = 1 by A13, A9, A5, A26, A27, Th37;
then (Upper_Seq (C,n)) /. 1 = (Gauge (C,n)) * (i,1) by A3, A4, PARTFUN1:def 6;
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:52
.= ((Gauge (C,n)) * (1,1)) `1 by A12, JORDAN1A:73 ;
hence contradiction by A1, A12, A11, GOBOARD5:3; :: thesis: verum

end;

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

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, A26, A27, Th38;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = (Gauge (C,n)) * (i,1) by A3, A4, PARTFUN1:def 6;
then A29: 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_1:46;
then (SE-corner (L~ (Cage (C,n)))) `2 < (E-max (L~ (Cage (C,n)))) `2 by SPRECT_2:53, XXREAL_0:2;
then S-bound (L~ (Cage (C,n))) < ((Gauge (C,n)) * (i,1)) `2 by A29, EUCLID:52;
hence contradiction by A1, JORDAN1A:72; :: thesis: verum

end;

end;