ex B being non empty strict subcategory of W -SUP_category st
( ( for a being object of (W -SUP_category ) holds
( a is object of B iff S₁[a] ) ) & ( for a, b being object of (W -SUP_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(A2, A3, A12);
hence ex b₁ being non empty strict subcategory of W -SUP_category st
( ( for a being object of (W -SUP_category ) holds a is object of b₁ ) & ( for a, b being object of (W -SUP_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 UpperAdj (@ f) is directed-sups-preserving ) ) ) ; :: thesis: verum