let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
set x = N-min (L~ (Cage (C,n)));
set p = W-min (L~ (Cage (C,n)));
set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:46;
A2: N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:39;
A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A4: 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;
A5: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A6: (Lower_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44;
then A7: (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (N-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, JORDAN1F:6, SPRECT_5:43;
(W-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A6, A3, JORDAN1F:6, SPRECT_5:42;
then N-min (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))) by A1, A2, A5, A3, A7, FINSEQ_6:62, XXREAL_0:2;
hence A8: N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th4; :: thesis: N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n))
len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def 8;
then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18;
hence N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A8; :: thesis: verum