let z be non constant standard clockwise_oriented special_circular_sequence; :: thesis: ( z /. 1 = E-max (L~ z) implies (S-max (L~ z)) .. z < (W-min (L~ z)) .. z )
assume A1: z /. 1 = E-max (L~ z) ; :: thesis: (S-max (L~ z)) .. z < (W-min (L~ z)) .. z
then A2: (S-max (L~ z)) .. z < (S-min (L~ z)) .. z by Th41;

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

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

hence (S-max (L~ z)) .. z < (W-min (L~ z)) .. z by A1, Th41; :: thesis: verum

end;

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

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

end;

end;