let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = E-min (L~ z) & S-min (L~ z) <> W-min (L~ z) implies (S-min (L~ z)) .. z < (W-min (L~ z)) .. z )
set g = Rotate z,(N-min (L~ z));
A1: L~ z = L~ (Rotate z,(N-min (L~ z))) by REVROT_1:33;
N-min (L~ z) in rng z by SPRECT_2:43;
then A2: (Rotate z,(N-min (L~ z))) /. 1 = N-min (L~ (Rotate z,(N-min (L~ z)))) by A1, FINSEQ_6:98;
then A3: ( W-min (L~ (Rotate z,(N-min (L~ z)))) in rng (Rotate z,(N-min (L~ z))) & (E-min (L~ (Rotate z,(N-min (L~ z))))) .. (Rotate z,(N-min (L~ z))) < (S-min (L~ (Rotate z,(N-min (L~ z))))) .. (Rotate z,(N-min (L~ z))) ) by Lm4, SPRECT_2:47;
assume that
A4: z /. 1 = E-min (L~ z) and
A5: S-min (L~ z) <> W-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,(N-min (L~ z))),(E-min (L~ z)) = z by A4, REVROT_1:16;
A7: ( S-min (L~ (Rotate z,(N-min (L~ z)))) in rng (Rotate z,(N-min (L~ z))) & E-min (L~ (Rotate z,(N-min (L~ z)))) in rng (Rotate z,(N-min (L~ z))) ) by SPRECT_2:45, SPRECT_2:49;
(S-min (L~ (Rotate z,(N-min (L~ z))))) .. (Rotate z,(N-min (L~ z))) < (W-min (L~ (Rotate z,(N-min (L~ z))))) .. (Rotate z,(N-min (L~ z))) by A1, A5, A2, SPRECT_2:78;
hence (S-min (L~ z)) .. z < (W-min (L~ z)) .. z by A1, A6, A7, A3, Th5; :: thesis: verum