let S be SubRelStr of L; :: thesis: ( S is finite-sups-inheriting implies ( S is bottom-inheriting & S is join-inheriting ) )
assume A1: S is finite-sups-inheriting ; :: thesis: ( S is bottom-inheriting & S is join-inheriting )
then ( ex_sup_of {} ,L implies "\/" (({} S),L) in the carrier of S ) ;
hence Bottom L in the carrier of S by YELLOW_0:42; :: according to WAYBEL34:def 19 :: thesis: S is join-inheriting
let x, y be Element of L; :: according to YELLOW_0:def 17 :: thesis: ( not x in the carrier of S or not y in the carrier of S or not ex_sup_of {x,y},L or "\/" ({x,y},L) in the carrier of S )
assume that
A2: x in the carrier of S and
A3: y in the carrier of S ; :: thesis: ( not ex_sup_of {x,y},L or "\/" ({x,y},L) in the carrier of S )
reconsider X = {x,y} as finite Subset of S by A2, A3, ZFMISC_1:32;
( ex_sup_of X,L implies "\/" (X,L) in the carrier of S ) by A1;
hence ( not ex_sup_of {x,y},L or "\/" ({x,y},L) in the carrier of S ) ; :: thesis: verum