let z be V19() standard clockwise_oriented special_circular_sequence; ( z /. 1 = S-max (L~ z) implies (N-min (L~ z)) .. z < (N-max (L~ z)) .. z )
set g = Rotate (z,(W-min (L~ z)));
A1:
L~ z = L~ (Rotate (z,(W-min (L~ z))))
by REVROT_1:33;
W-min (L~ z) in rng z
by SPRECT_2:43;
then
(Rotate (z,(W-min (L~ z)))) /. 1 = W-min (L~ (Rotate (z,(W-min (L~ z)))))
by A1, FINSEQ_6:92;
then A2:
( (N-min (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) < (N-max (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) & (N-max (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) < (S-max (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) )
by Lm14, Th24;
A3:
S-max (L~ (Rotate (z,(W-min (L~ z))))) in rng (Rotate (z,(W-min (L~ z))))
by SPRECT_2:42;
assume A4:
z /. 1 = S-max (L~ z)
; (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 A5:
Rotate ((Rotate (z,(W-min (L~ z)))),(S-max (L~ z))) = z
by A4, FINSEQ_6:181;
( N-max (L~ (Rotate (z,(W-min (L~ z))))) in rng (Rotate (z,(W-min (L~ z)))) & N-min (L~ (Rotate (z,(W-min (L~ z))))) in rng (Rotate (z,(W-min (L~ z)))) )
by SPRECT_2:39, SPRECT_2:40;
hence
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
by A1, A5, A3, A2, Th11; verum