reconsider x9 = x as Element of L ;
thus not downarrow x is empty ; :: thesis: downarrow x is directed
let z, y be Element of L; :: according to WAYBEL_0:def 1 :: thesis: ( z in downarrow x & y in downarrow x implies ex z being Element of L st
( z in downarrow x & z <= z & y <= z ) )
assume that
A1: z in downarrow x and
A2: y in downarrow x ; :: thesis: ex z being Element of L st
( z in downarrow x & z <= z & y <= z )
take x9 ; :: thesis: ( x9 in downarrow x & z <= x9 & y <= x9 )
x9 <= x9 ;
hence ( x9 in downarrow x & z <= x9 & y <= x9 ) by A1, A2, Th17; :: thesis: verum