RelStr(# a,r #) is object of (W -SUP_category) by A1, Def5;
then reconsider b = RelStr(# a,r #) as object of (W -SUP(SO)_category) by Def11;
b = latt b ;
then A2: ex b being object of (W -SUP(SO)_category) st S₁[b] ;
thus ex B being non empty strict full subcategory of W -SUP(SO)_category st
for a being object of (W -SUP(SO)_category) holds
( a is object of B iff S₁[a] ) from YELLOW20:sch 2(A2); :: thesis: verum