let n be Nat; for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
set Wmi = W-min (L~ (Cage (C,n)));
set Wma = W-max (L~ (Cage (C,n)));
set Nmi = N-min (L~ (Cage (C,n)));
set Nma = N-max (L~ (Cage (C,n)));
set Ema = E-max (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1:
L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n))
by REVROT_1:33;
then A2:
W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))
by SPRECT_2:43;
A3:
W-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))
by A1, SPRECT_2:44;
A4:
E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))
by A1, SPRECT_2:46;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n))
by SPRECT_2:43;
then A5:
(Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n)))
by FINSEQ_6:92;
then A6:
(W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))
by A1, SPRECT_5:21;
A7:
( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (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 A1, A5, JORDAN1E:def 1, SPRECT_5:25;
(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))))))
by A1, A5, SPRECT_5:24;
then A8:
(W-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))))))
by A1, A5, SPRECT_5:23, XXREAL_0:2;
then
(W-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 A1, A5, SPRECT_5:25, XXREAL_0:2;
then
(W-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))
by A2, A4, A6, SPRECT_5:3, XXREAL_0:2;
hence
(W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
by A4, A6, A7, A8, A3, SPRECT_5:3, XXREAL_0:2; verum