let L be antisymmetric lower-bounded with_infima RelStr ; :: thesis: for X being non empty Subset of L holds X "/\" {(Bottom L)} = {(Bottom L)}

let X be non empty Subset of L; :: thesis: X "/\" {(Bottom L)} = {(Bottom L)}

thus X "/\" {(Bottom L)} c= {(Bottom L)} by Th10; :: according to XBOOLE_0:def 10 :: thesis: {(Bottom L)} c= X "/\" {(Bottom L)}

let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in {(Bottom L)} or q in X "/\" {(Bottom L)} )

assume q in {(Bottom L)} ; :: thesis: q in X "/\" {(Bottom L)}

then A1: ( X "/\" {(Bottom L)} = { ((Bottom L) "/\" y) where y is Element of L : y in X } & q = Bottom L ) by TARSKI:def 1, YELLOW_4:42;

consider y being object such that

A2: y in X by XBOOLE_0:def 1;

reconsider y = y as Element of X by A2;

(Bottom L) "/\" y = Bottom L by WAYBEL_1:3;

hence q in X "/\" {(Bottom L)} by A1; :: thesis: verum

let X be non empty Subset of L; :: thesis: X "/\" {(Bottom L)} = {(Bottom L)}

thus X "/\" {(Bottom L)} c= {(Bottom L)} by Th10; :: according to XBOOLE_0:def 10 :: thesis: {(Bottom L)} c= X "/\" {(Bottom L)}

let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in {(Bottom L)} or q in X "/\" {(Bottom L)} )

assume q in {(Bottom L)} ; :: thesis: q in X "/\" {(Bottom L)}

then A1: ( X "/\" {(Bottom L)} = { ((Bottom L) "/\" y) where y is Element of L : y in X } & q = Bottom L ) by TARSKI:def 1, YELLOW_4:42;

consider y being object such that

A2: y in X by XBOOLE_0:def 1;

reconsider y = y as Element of X by A2;

(Bottom L) "/\" y = Bottom L by WAYBEL_1:3;

hence q in X "/\" {(Bottom L)} by A1; :: thesis: verum