let L be antisymmetric upper-bounded with_suprema RelStr ; :: thesis: for X being non empty Subset of L holds X "\/" {(Top L)} = {(Top L)}
let X be non empty Subset of L; :: thesis: X "\/" {(Top L)} = {(Top L)}
A1: X "\/" {(Top L)} = { ((Top L) "\/" y) where y is Element of L : y in X } by YELLOW_4:15;
thus X "\/" {(Top L)} c= {(Top L)} by Th16; :: according to XBOOLE_0:def 10 :: thesis: {(Top L)} c= X "\/" {(Top L)}
let q be set ; :: according to TARSKI:def 3 :: thesis: ( not q in {(Top L)} or q in X "\/" {(Top L)} )
assume q in {(Top L)} ; :: thesis: q in X "\/" {(Top L)}
then A2: q = Top L by TARSKI:def 1;
consider y being set such that
A3: y in X by XBOOLE_0:def 1;
reconsider y = y as Element of X by A3;
(Top L) "\/" y = Top L by WAYBEL_1:5;
hence q in X "\/" {(Top L)} by A1, A2; :: thesis: verum