let z be constant standard clockwise_oriented special_circular_sequence; ( z /. 1 = E-max (L~ z) & N-max (L~ z) <> E-max (L~ z) implies (N-min (L~ z)) .. z < (N-max (L~ z)) .. z )
set g = Rotate (z,(S-max (L~ z)));
A1:
L~ z = L~ (Rotate (z,(S-max (L~ z))))
by REVROT_1:33;
assume A2:
z /. 1 = E-max (L~ z)
; ( not N-max (L~ z) <> E-max (L~ z) or (N-min (L~ z)) .. z < (N-max (L~ z)) .. z )
for i being Nat st 1 < i & i < len z holds
z /. i <> z /. 1
by GOBOARD7:36;
then A3:
Rotate ((Rotate (z,(S-max (L~ z)))),(E-max (L~ z))) = z
by A2, FINSEQ_6:181;
A4:
( N-max (L~ (Rotate (z,(S-max (L~ z))))) in rng (Rotate (z,(S-max (L~ z)))) & N-min (L~ (Rotate (z,(S-max (L~ z))))) in rng (Rotate (z,(S-max (L~ z)))) )
by SPRECT_2:39, SPRECT_2:40;
S-max (L~ z) in rng z
by SPRECT_2:42;
then A5:
(Rotate (z,(S-max (L~ z)))) /. 1 = S-max (L~ (Rotate (z,(S-max (L~ z)))))
by A1, FINSEQ_6:92;
then A6:
( E-max (L~ (Rotate (z,(S-max (L~ z))))) in rng (Rotate (z,(S-max (L~ z)))) & (N-min (L~ (Rotate (z,(S-max (L~ z)))))) .. (Rotate (z,(S-max (L~ z)))) < (N-max (L~ (Rotate (z,(S-max (L~ z)))))) .. (Rotate (z,(S-max (L~ z)))) )
by Th34, SPRECT_2:46;
assume
N-max (L~ z) <> E-max (L~ z)
; (N-min (L~ z)) .. z < (N-max (L~ z)) .. z
then
(N-max (L~ (Rotate (z,(S-max (L~ z)))))) .. (Rotate (z,(S-max (L~ z)))) < (E-max (L~ (Rotate (z,(S-max (L~ z)))))) .. (Rotate (z,(S-max (L~ z))))
by A1, A5, Th35;
hence
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
by A1, A3, A4, A6, Th11; verum