let L be non empty Poset; :: thesis: for x being Element of L holds

( ex_inf_of uparrow x,L & inf (uparrow x) = x )

let x be Element of L; :: thesis: ( ex_inf_of uparrow x,L & inf (uparrow x) = x )

ex_inf_of {x},L by YELLOW_0:38;

hence ex_inf_of uparrow x,L by Th37; :: thesis: inf (uparrow x) = x

thus inf (uparrow x) = inf {x} by Th38, YELLOW_0:38

.= x by YELLOW_0:39 ; :: thesis: verum

( ex_inf_of uparrow x,L & inf (uparrow x) = x )

let x be Element of L; :: thesis: ( ex_inf_of uparrow x,L & inf (uparrow x) = x )

ex_inf_of {x},L by YELLOW_0:38;

hence ex_inf_of uparrow x,L by Th37; :: thesis: inf (uparrow x) = x

thus inf (uparrow x) = inf {x} by Th38, YELLOW_0:38

.= x by YELLOW_0:39 ; :: thesis: verum