let B1, B2 be non empty strict subcategory of W -SUP_category ; :: thesis: ( ( 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 ) ) & ( for a being Object of (W -SUP_category) holds a is Object of B2 ) & ( for a, b being Object of (W -SUP_category)
for a9, b9 being Object of B2 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 ) ) implies B1 = B2 )
assume for a being Object of (W -SUP_category) holds a is Object of B1 ; :: thesis: ( ex a, b being Object of (W -SUP_category) ex a9, b9 being Object of B1 st
( a9 = a & b9 = b & <^a,b^> <> {} & not for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) or ex a being Object of (W -SUP_category) st a is not Object of B2 or ex a, b being Object of (W -SUP_category) ex a9, b9 being Object of B2 st
( a9 = a & b9 = b & <^a,b^> <> {} & not for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) or B1 = B2 )
then A14: for a being Object of (W -SUP_category) holds
( a is Object of B1 iff S₁[a] ) ;
assume A15: 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 S₂[a,b,f] ) ; :: thesis: ( ex a being Object of (W -SUP_category) st a is not Object of B2 or ex a, b being Object of (W -SUP_category) ex a9, b9 being Object of B2 st
( a9 = a & b9 = b & <^a,b^> <> {} & not for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) or B1 = B2 )
assume for a being Object of (W -SUP_category) holds a is Object of B2 ; :: thesis: ( ex a, b being Object of (W -SUP_category) ex a9, b9 being Object of B2 st
( a9 = a & b9 = b & <^a,b^> <> {} & not for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) or B1 = B2 )
then A16: for a being Object of (W -SUP_category) holds
( a is Object of B2 iff S₁[a] ) ;
assume A17: for a, b being Object of (W -SUP_category)
for a9, b9 being Object of B2 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] ) ; :: thesis: B1 = B2
AltCatStr(# the carrier of B1, the Arrows of B1, the Comp of B1 #) = AltCatStr(# the carrier of B2, the Arrows of B2, the Comp of B2 #) from YELLOW20:sch 1(A14, A15, A16, A17);
hence B1 = B2 ; :: thesis: verum