let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = E-max (L~ z) implies (S-min (L~ z)) .. z < (W-max (L~ z)) .. z )
assume A1: z /. 1 = E-max (L~ z) ; :: thesis: (S-min (L~ z)) .. z < (W-max (L~ z)) .. z
then A2: (W-min (L~ z)) .. z < (W-max (L~ z)) .. z by Th43;

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

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

hence (S-min (L~ z)) .. z < (W-max (L~ z)) .. z by A1, Th43; :: thesis: verum

end;

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

thus (S-min (L~ z)) .. z < (W-max (L~ z)) .. z by A1, A2, Th42, XXREAL_0:2; :: thesis: verum

end;

end;