let h be non constant standard special_circular_sequence; :: thesis: n_s_w h <> n_s_e h

set i1 = n_s_w h;

set i2 = n_s_e h;

A1: n_s_w h <= (n_s_w h) + 1 by NAT_1:11;

A2: 1 <= n_s_w h by Def9;

(n_s_w h) + 1 <= len h by Def9;

then n_s_w h <= len h by A1, XXREAL_0:2;

then n_s_w h in dom h by A2, FINSEQ_3:25;

then A3: h . (n_s_w h) = h /. (n_s_w h) by PARTFUN1:def 6;

A4: n_s_e h <= (n_s_e h) + 1 by NAT_1:11;

A5: h . (n_s_e h) = E-min (L~ h) by Def11;

A6: 1 <= n_s_e h by Def11;

(n_s_e h) + 1 <= len h by Def11;

then n_s_e h <= len h by A4, XXREAL_0:2;

then n_s_e h in dom h by A6, FINSEQ_3:25;

then A7: h . (n_s_e h) = h /. (n_s_e h) by PARTFUN1:def 6;

A8: h . (n_s_w h) = W-min (L~ h) by Def9;

thus n_s_w h <> n_s_e h :: thesis: verum

set i1 = n_s_w h;

set i2 = n_s_e h;

A1: n_s_w h <= (n_s_w h) + 1 by NAT_1:11;

A2: 1 <= n_s_w h by Def9;

(n_s_w h) + 1 <= len h by Def9;

then n_s_w h <= len h by A1, XXREAL_0:2;

then n_s_w h in dom h by A2, FINSEQ_3:25;

then A3: h . (n_s_w h) = h /. (n_s_w h) by PARTFUN1:def 6;

A4: n_s_e h <= (n_s_e h) + 1 by NAT_1:11;

A5: h . (n_s_e h) = E-min (L~ h) by Def11;

A6: 1 <= n_s_e h by Def11;

(n_s_e h) + 1 <= len h by Def11;

then n_s_e h <= len h by A4, XXREAL_0:2;

then n_s_e h in dom h by A6, FINSEQ_3:25;

then A7: h . (n_s_e h) = h /. (n_s_e h) by PARTFUN1:def 6;

A8: h . (n_s_w h) = W-min (L~ h) by Def9;

thus n_s_w h <> n_s_e h :: thesis: verum

proof

assume
n_s_w h = n_s_e h
; :: thesis: contradiction

then A9: W-bound (L~ h) = (h /. (n_s_e h)) `1 by A8, A3, EUCLID:52

.= E-bound (L~ h) by A5, A7, EUCLID:52 ;

A10: 1 <= len h by GOBOARD7:34, XXREAL_0:2;

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 A12, FINSEQ_3:25;

A15: 1 <= ii by A12, FINSEQ_3:25;

then A16: (h /. ii) `1 <= E-bound (L~ h) by A14, Th12;

A17: (h /. ii) `1 >= W-bound (L~ h) by A15, A14, Th12;

(h /. 1) `1 <= E-bound (L~ h) by A10, Th12;

then (h /. 1) `1 = W-bound (L~ h) by A9, A11, XXREAL_0:1;

hence contradiction by A9, A13, A16, A17, XXREAL_0:1; :: thesis: verum

end;then A9: W-bound (L~ h) = (h /. (n_s_e h)) `1 by A8, A3, EUCLID:52

.= E-bound (L~ h) by A5, A7, EUCLID:52 ;

A10: 1 <= len h by GOBOARD7:34, XXREAL_0:2;

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 A12, FINSEQ_3:25;

A15: 1 <= ii by A12, FINSEQ_3:25;

then A16: (h /. ii) `1 <= E-bound (L~ h) by A14, Th12;

A17: (h /. ii) `1 >= W-bound (L~ h) by A15, A14, Th12;

(h /. 1) `1 <= E-bound (L~ h) by A10, Th12;

then (h /. 1) `1 = W-bound (L~ h) by A9, A11, XXREAL_0:1;

hence contradiction by A9, A13, A16, A17, XXREAL_0:1; :: thesis: verum