let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n)))

let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n)))

set Ca = Cage (C,n);

set LS = Lower_Seq (C,n);

set Emax = E-max (L~ (Cage (C,n)));

set Emin = E-min (L~ (Cage (C,n)));

set Smax = S-max (L~ (Cage (C,n)));

set Smin = S-min (L~ (Cage (C,n)));

set Wmin = W-min (L~ (Cage (C,n)));

set Nmin = N-min (L~ (Cage (C,n)));

W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;

then A1: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46;

len (Lower_Seq (C,n)) >= 3 by JORDAN1E:15;

then len (Lower_Seq (C,n)) >= 2 by XXREAL_0:2;

then 2 <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by Th30;

then 2 <= (W-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) by Th18;

then 2 <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A1, FINSEQ_6:72;

then A2: 2 in Seg ((W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by FINSEQ_1:1;

((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53;

then A3: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42;

( N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:11;

then (N-max (L~ (Cage (C,n)))) `1 <= (E-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24;

then N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by SPRECT_2:51;

then A4: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57;

A5: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:71;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:72, XXREAL_0:2;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:73, XXREAL_0:2;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:74, XXREAL_0:2;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A5, SPRECT_2:76, XXREAL_0:2;

then A6: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;

A7: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;

then A8: 1 <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by FINSEQ_4:21;

((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A7, FINSEQ_5:54

.= (Cage (C,n)) /. 1 by FINSEQ_6:def 1

.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;

then A9: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:168;

{(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A9, A3, TARSKI:def 2;

then A10: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_1:11;

card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_2:61;

then card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by CARD_1:62;

then Segm 2 c= Segm (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A4, A10;

then A11: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39;

then A12: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 1 by XXREAL_0:2;

A13: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 1 by Th18

.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A1, FINSEQ_5:44

.= (Cage (C,n)) /. ((1 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A12, FINSEQ_6:174

.= (Cage (C,n)) /. (0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_1:232 ;

(Lower_Seq (C,n)) /. 2 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 2 by Th18

.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 2 by A1, A2, FINSEQ_5:43

.= (Cage (C,n)) /. ((2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A11, FINSEQ_6:174

.= (Cage (C,n)) /. ((2 - 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_0:def 2 ;

hence ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n))) by A8, A6, A13, JORDAN1E:20, JORDAN1F:6; :: thesis: verum

let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n)))

set Ca = Cage (C,n);

set LS = Lower_Seq (C,n);

set Emax = E-max (L~ (Cage (C,n)));

set Emin = E-min (L~ (Cage (C,n)));

set Smax = S-max (L~ (Cage (C,n)));

set Smin = S-min (L~ (Cage (C,n)));

set Wmin = W-min (L~ (Cage (C,n)));

set Nmin = N-min (L~ (Cage (C,n)));

W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;

then A1: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46;

len (Lower_Seq (C,n)) >= 3 by JORDAN1E:15;

then len (Lower_Seq (C,n)) >= 2 by XXREAL_0:2;

then 2 <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by Th30;

then 2 <= (W-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) by Th18;

then 2 <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A1, FINSEQ_6:72;

then A2: 2 in Seg ((W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by FINSEQ_1:1;

((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53;

then A3: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42;

( N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:11;

then (N-max (L~ (Cage (C,n)))) `1 <= (E-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24;

then N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by SPRECT_2:51;

then A4: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57;

A5: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:71;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:72, XXREAL_0:2;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:73, XXREAL_0:2;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:74, XXREAL_0:2;

then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A5, SPRECT_2:76, XXREAL_0:2;

then A6: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;

A7: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;

then A8: 1 <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by FINSEQ_4:21;

((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A7, FINSEQ_5:54

.= (Cage (C,n)) /. 1 by FINSEQ_6:def 1

.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;

then A9: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:168;

{(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A9, A3, TARSKI:def 2;

then A10: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_1:11;

card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_2:61;

then card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by CARD_1:62;

then Segm 2 c= Segm (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A4, A10;

then A11: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39;

then A12: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 1 by XXREAL_0:2;

A13: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 1 by Th18

.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A1, FINSEQ_5:44

.= (Cage (C,n)) /. ((1 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A12, FINSEQ_6:174

.= (Cage (C,n)) /. (0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_1:232 ;

(Lower_Seq (C,n)) /. 2 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 2 by Th18

.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 2 by A1, A2, FINSEQ_5:43

.= (Cage (C,n)) /. ((2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A11, FINSEQ_6:174

.= (Cage (C,n)) /. ((2 - 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_0:def 2 ;

hence ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n))) by A8, A6, A13, JORDAN1E:20, JORDAN1F:6; :: thesis: verum