let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
set Ema = E-max (L~ (Cage (C,n)));
set Emi = E-min (L~ (Cage (C,n)));
set Sma = S-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: E-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:45;
A4: S-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:42;
A5: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A6: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92;
then A7: (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_5:39;
A8: (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, A6, SPRECT_5:41;
A9: (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:40;
then (S-max (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, A6, SPRECT_5:41, XXREAL_0:2;
then (E-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A5, A7, SPRECT_5:3, XXREAL_0:2;
hence (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A5, A7, A9, A8, A4, SPRECT_5:3, XXREAL_0:2; :: thesis: verum