let f be Function of L1,L2; :: thesis: ( f is sups-preserving implies ( f is directed-sups-preserving & f is join-preserving ) )
assume A2: for X being Subset of L1 holds f preserves_sup_of X ; :: according to WAYBEL_0:def 33 :: thesis: ( f is directed-sups-preserving & f is join-preserving )
hence for X being Subset of L1 st not X is empty & X is directed holds
f preserves_sup_of X ; :: according to WAYBEL_0:def 37 :: thesis: f is join-preserving
thus for x, y being Element of L1 holds f preserves_sup_of {x,y} by A2; :: according to WAYBEL_0:def 35 :: thesis: verum