let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = S-min (L~ z) & W-max (L~ z) <> N-min (L~ z) implies (W-max (L~ z)) .. z < (N-min (L~ z)) .. z )
set g = Rotate z,(E-max (L~ z));
A1: for i being Element of NAT st 1 < i & i < len z holds
z /. i <> z /. 1 by GOBOARD7:38;
A2: E-max (L~ z) in rng z by SPRECT_2:50;
A3: L~ z = L~ (Rotate z,(E-max (L~ z))) by REVROT_1:33;
assume that
A4: z /. 1 = S-min (L~ z) and
A5: W-max (L~ z) <> N-min (L~ z) ; :: thesis: (W-max (L~ z)) .. z < (N-min (L~ z)) .. z
A6: Rotate (Rotate z,(E-max (L~ z))),(S-min (L~ z)) = z by A1, A4, REVROT_1:16;
A7: (Rotate z,(E-max (L~ z))) /. 1 = E-max (L~ (Rotate z,(E-max (L~ z)))) by A2, A3, FINSEQ_6:98;
A8: W-max (L~ (Rotate z,(E-max (L~ z)))) in rng (Rotate z,(E-max (L~ z))) by SPRECT_2:48;
A9: S-min (L~ (Rotate z,(E-max (L~ z)))) in rng (Rotate z,(E-max (L~ z))) by SPRECT_2:45;
A10: N-min (L~ (Rotate z,(E-max (L~ z)))) in rng (Rotate z,(E-max (L~ z))) by SPRECT_2:43;
A11: (S-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 A7, Lm30;
(W-max (L~ (Rotate z,(E-max (L~ z))))) .. (Rotate z,(E-max (L~ z))) < (N-min (L~ (Rotate z,(E-max (L~ z))))) .. (Rotate z,(E-max (L~ z))) by A3, A5, A7, Th44;
hence (W-max (L~ z)) .. z < (N-min (L~ z)) .. z by A3, A6, A8, A9, A10, A11, Th5; :: thesis: verum