let n be Nat; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
set Ema = E-max (L~ (Cage (C,n)));
set Smi = S-min (L~ (Cage (C,n)));
set Wmi = W-min (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1:
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))
by Th18;
A2:
L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n))
by REVROT_1:33;
then A3:
W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))
by SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n))
by SPRECT_2:46;
then
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n)))
by FINSEQ_6:92;
then A4:
(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 A2, SPRECT_5:41;
S-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))
by A2, SPRECT_2:41;
then
(S-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))
by A3, A4, SPRECT_5:3;
hence
(S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
by A1, A3, A4, SPRECT_5:3; verum