let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = S-max (L~ z) implies (W-min (L~ z)) .. z < (W-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_2:47;
then (Rotate (z,(W-min (L~ z)))) /. 1 = W-min (L~ (Rotate (z,(W-min (L~ z))))) by A1, FINSEQ_6:98;
then A2: ( (S-max (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) > (W-max (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) & (W-min (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) < (W-max (L~ (Rotate (z,(W-min (L~ z)))))) .. (Rotate (z,(W-min (L~ z)))) ) by Lm15, Th22;
A3: W-min (L~ (Rotate (z,(W-min (L~ z))))) in rng (Rotate (z,(W-min (L~ z)))) by SPRECT_2:47;
assume A4: z /. 1 = S-max (L~ z) ; :: thesis: (W-min (L~ z)) .. z < (W-max (L~ z)) .. z
for i being Element of NAT st 1 < i & i < len z holds
z /. i <> z /. 1 by GOBOARD7:38;
then A5: Rotate ((Rotate (z,(W-min (L~ z)))),(S-max (L~ z))) = z by A4, REVROT_1:16;
( S-max (L~ (Rotate (z,(W-min (L~ z))))) in rng (Rotate (z,(W-min (L~ z)))) & W-max (L~ (Rotate (z,(W-min (L~ z))))) in rng (Rotate (z,(W-min (L~ z)))) ) by SPRECT_2:46, SPRECT_2:48;
hence (W-min (L~ z)) .. z < (W-max (L~ z)) .. z by A1, A5, A3, A2, Th12; :: thesis: verum