let L be non empty Poset; :: thesis: for x being Element of L holds
( ex_sup_of downarrow x,L & sup (downarrow x) = x )
let x be Element of L; :: thesis: ( ex_sup_of downarrow x,L & sup (downarrow x) = x )
ex_sup_of {x},L by YELLOW_0:38;
hence ex_sup_of downarrow x,L by Th32; :: thesis: sup (downarrow x) = x
thus sup (downarrow x) = sup {x} by Th33, YELLOW_0:38
.= x by YELLOW_0:39 ; :: thesis: verum