let f be non constant standard special_circular_sequence; :: thesis: ( f /. 1 = N-min (L~ f) implies (N-min (L~ f)) .. f < (N-max (L~ f)) .. f )
A1: N-min (L~ f) in rng f by Th43;
A2: N-max (L~ f) in rng f by Th44;
then A3: (N-min (L~ f)) .. f <> (N-max (L~ f)) .. f by A1, Th56, FINSEQ_5:10;
(N-max (L~ f)) .. f in dom f by A2, FINSEQ_4:30;
then A4: (N-max (L~ f)) .. f >= 1 by FINSEQ_3:27;
assume f /. 1 = N-min (L~ f) ; :: thesis: (N-min (L~ f)) .. f < (N-max (L~ f)) .. f
then (N-min (L~ f)) .. f = 1 by FINSEQ_6:47;
hence (N-min (L~ f)) .. f < (N-max (L~ f)) .. f by A3, A4, XXREAL_0:1; :: thesis: verum