let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = S-max (L~ z) implies (W-min (L~ z)) .. z < (N-min (L~ z)) .. z )
assume A1: z /. 1 = S-max (L~ z) ; :: thesis: (W-min (L~ z)) .. z < (N-min (L~ z)) .. z
then A2: (W-min (L~ z)) .. z < (W-max (L~ z)) .. z by Th33;

per cases ( N-min (L~ z) = W-max (L~ z) or N-min (L~ z) <> W-max (L~ z) ) ;

suppose N-min (L~ z) = W-max (L~ z) ; :: thesis: (W-min (L~ z)) .. z < (N-min (L~ z)) .. z

hence (W-min (L~ z)) .. z < (N-min (L~ z)) .. z by A1, Th33; :: thesis: verum

end;

suppose N-min (L~ z) <> W-max (L~ z) ; :: thesis: (W-min (L~ z)) .. z < (N-min (L~ z)) .. z

thus (W-min (L~ z)) .. z < (N-min (L~ z)) .. z by A1, A2, Th34, XXREAL_0:2; :: thesis: verum

end;

end;