let L be non empty RelStr ; for S being Subset of L holds
( S is sups-closed iff for X being Subset of S st ex_sup_of X,L holds
"\/" (X,L) in S )
let S be Subset of L; ( S is sups-closed iff for X being Subset of S st ex_sup_of X,L holds
"\/" (X,L) in S )
thus
( S is sups-closed implies for X being Subset of S st ex_sup_of X,L holds
"\/" (X,L) in S )
( ( for X being Subset of S st ex_sup_of X,L holds
"\/" (X,L) in S ) implies S is sups-closed )
assume A3:
for X being Subset of S st ex_sup_of X,L holds
"\/" (X,L) in S
; S is sups-closed
then
subrelstr S is sups-inheriting
;
hence
S is sups-closed
; verum