let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = W-min (L~ z) implies (N-max (L~ z)) .. z < (S-max (L~ z)) .. z )
assume A1: z /. 1 = W-min (L~ z) ; :: thesis: (N-max (L~ z)) .. z < (S-max (L~ z)) .. z
then A2: (E-max (L~ z)) .. z < (S-max (L~ z)) .. z by Lm11;

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

suppose N-max (L~ z) = E-max (L~ z) ; :: thesis: (N-max (L~ z)) .. z < (S-max (L~ z)) .. z

hence (N-max (L~ z)) .. z < (S-max (L~ z)) .. z by A1, Lm11; :: thesis: verum

end;

suppose N-max (L~ z) <> E-max (L~ z) ; :: thesis: (N-max (L~ z)) .. z < (S-max (L~ z)) .. z

thus (N-max (L~ z)) .. z < (S-max (L~ z)) .. z by A1, A2, Th26, XXREAL_0:2; :: thesis: verum

end;

end;