ex B being non empty strict subcategory of W -INF_category st
( ( for a being Object of (W -INF_category) holds
( a is Object of B iff S₁[a] ) ) & ( for a, b being Object of (W -INF_category)
for a9, b9 being Object of B st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff S₂[a,b,f] ) ) ) from YELLOW18:sch 7(A1, A2, A8);
hence ex b₁ being non empty strict subcategory of W -INF_category st
( ( for a being Object of (W -INF_category) holds a is Object of b₁ ) & ( for a, b being Object of (W -INF_category)
for a9, b9 being Object of b₁ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff @ f is directed-sups-preserving ) ) ) ; :: thesis: verum