let L be non empty antisymmetric lower-bounded RelStr ; :: thesis: for X being non empty Subset of L holds Bottom L in downarrow X

let X be non empty Subset of L; :: thesis: Bottom L in downarrow X

consider y being object such that

A1: y in X by XBOOLE_0:def 1;

reconsider y = y as Element of X by A1;

( downarrow X = { x where x is Element of L : ex y being Element of L st

( x <= y & y in X ) } & Bottom L <= y ) by WAYBEL_0:14, YELLOW_0:44;

hence Bottom L in downarrow X ; :: thesis: verum

let X be non empty Subset of L; :: thesis: Bottom L in downarrow X

consider y being object such that

A1: y in X by XBOOLE_0:def 1;

reconsider y = y as Element of X by A1;

( downarrow X = { x where x is Element of L : ex y being Element of L st

( x <= y & y in X ) } & Bottom L <= y ) by WAYBEL_0:14, YELLOW_0:44;

hence Bottom L in downarrow X ; :: thesis: verum