let h be non constant standard special_circular_sequence; :: thesis:
set i1 = n_n_w h;
set i2 = n_n_e h;
A1: n_n_w h <= () + 1 by NAT_1:11;
A2: 1 <= n_n_w h by Def10;
(n_n_w h) + 1 <= len h by Def10;
then n_n_w h <= len h by ;
then n_n_w h in dom h by ;
then A3: h . () = h /. () by PARTFUN1:def 6;
A4: n_n_e h <= () + 1 by NAT_1:11;
A5: h . () = E-max (L~ h) by Def12;
A6: 1 <= n_n_e h by Def12;
(n_n_e h) + 1 <= len h by Def12;
then n_n_e h <= len h by ;
then n_n_e h in dom h by ;
then A7: h . () = h /. () by PARTFUN1:def 6;
A8: h . () = W-max (L~ h) by Def10;
thus n_n_w h <> n_n_e h :: thesis: verum
proof
assume n_n_w h = n_n_e h ; :: thesis: contradiction
then A9: W-bound (L~ h) = (h /. ()) `1 by
.= E-bound (L~ h) by ;
A10: 1 <= len h by ;
then A11: (h /. 1) `1 >= W-bound (L~ h) by Th12;
consider ii being Nat such that
A12: ii in dom h and
A13: (h /. ii) `1 <> (h /. 1) `1 by GOBOARD7:30;
A14: ii <= len h by ;
A15: 1 <= ii by ;
then A16: (h /. ii) `1 <= E-bound (L~ h) by ;
A17: (h /. ii) `1 >= W-bound (L~ h) by ;
(h /. 1) `1 <= E-bound (L~ h) by ;
then (h /. 1) `1 = W-bound (L~ h) by ;
hence contradiction by A9, A13, A16, A17, XXREAL_0:1; :: thesis: verum
end;