let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = W-max (L~ z) & W-min (L~ z) <> S-min (L~ z) implies (S-min (L~ z)) .. z < (W-min (L~ z)) .. z )
set g = Rotate (z,(E-max (L~ z)));
A1: L~ z = L~ (Rotate (z,(E-max (L~ z)))) by REVROT_1:33;
E-max (L~ z) in rng z by SPRECT_2:50;
then A2: (Rotate (z,(E-max (L~ z)))) /. 1 = E-max (L~ (Rotate (z,(E-max (L~ z))))) by A1, FINSEQ_6:98;
then A3: (W-min (L~ (Rotate (z,(E-max (L~ z)))))) .. (Rotate (z,(E-max (L~ z)))) < (W-max (L~ (Rotate (z,(E-max (L~ z)))))) .. (Rotate (z,(E-max (L~ z)))) by Th43;
assume that
A4: z /. 1 = W-max (L~ z) and
A5: W-min (L~ z) <> S-min (L~ z) ; :: thesis: (S-min (L~ z)) .. z < (W-min (L~ z)) .. z
for i being Element of NAT st 1 < i & i < len z holds
z /. i <> z /. 1 by GOBOARD7:38;
then A6: Rotate ((Rotate (z,(E-max (L~ z)))),(W-max (L~ z))) = z by A4, REVROT_1:16;
A7: ( W-min (L~ (Rotate (z,(E-max (L~ z))))) in rng (Rotate (z,(E-max (L~ z)))) & S-min (L~ (Rotate (z,(E-max (L~ z))))) in rng (Rotate (z,(E-max (L~ z)))) ) by SPRECT_2:45, SPRECT_2:47;
( W-max (L~ (Rotate (z,(E-max (L~ z))))) in rng (Rotate (z,(E-max (L~ z)))) & (S-min (L~ (Rotate (z,(E-max (L~ z)))))) .. (Rotate (z,(E-max (L~ z)))) < (W-min (L~ (Rotate (z,(E-max (L~ z)))))) .. (Rotate (z,(E-max (L~ z)))) ) by A1, A5, A2, Th42, SPRECT_2:48;
hence (S-min (L~ z)) .. z < (W-min (L~ z)) .. z by A1, A6, A7, A3, Th12; :: thesis: verum