let f be non constant standard special_circular_sequence; :: thesis: ( f /. 1 = N-min (L~ f) implies (N-min (L~ f)) .. f < (E-max (L~ f)) .. f )
A1: E-max (L~ f) in rng f by Th50;
A2: N-min (L~ f) in rng f by Th43;
(N-max (L~ f)) `1 <= (NE-corner (L~ f)) `1 by PSCOMP_1:97;
then (N-max (L~ f)) `1 <= E-bound (L~ f) by EUCLID:56;
then (N-min (L~ f)) `1 < E-bound (L~ f) by Th55, XXREAL_0:2;
then (N-min (L~ f)) `1 < (E-max (L~ f)) `1 by EUCLID:56;
then A3: (N-min (L~ f)) .. f <> (E-max (L~ f)) .. f by A1, A2, FINSEQ_5:10;
assume f /. 1 = N-min (L~ f) ; :: thesis: (N-min (L~ f)) .. f < (E-max (L~ f)) .. f
then A4: (N-min (L~ f)) .. f = 1 by FINSEQ_6:47;
(E-max (L~ f)) .. f >= 1 by A1, FINSEQ_4:31;
hence (N-min (L~ f)) .. f < (E-max (L~ f)) .. f by A3, A4, XXREAL_0:1; :: thesis: verum