let L be non empty reflexive RelStr ; :: thesis: for x being set holds
( x is Element of (DsupClOpers L) iff x is directed-sups-preserving closure Function of L,L )
let x be set ; :: thesis: ( x is Element of (DsupClOpers L) iff x is directed-sups-preserving closure Function of L,L )
( x is Element of (ClOpers L) iff x is closure Function of L,L ) by Th10;
hence ( x is Element of (DsupClOpers L) iff x is directed-sups-preserving closure Function of L,L ) by Def6, YELLOW_0:58; :: thesis: verum