let T be reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure ; :: thesis: for t being type of T
for A being Subset of the adjectives of T st A is_applicable_to t holds
A ast t in types A
let t be type of T; :: thesis: for A being Subset of the adjectives of T st A is_applicable_to t holds
A ast t in types A
let a be Subset of the adjectives of T; :: thesis: ( a is_applicable_to t implies a ast t in types a )
assume a is_applicable_to t ; :: thesis: a ast t in types a
then (types a) /\ (downarrow t) is Ideal of T by Th26;
then sup ((types a) /\ (downarrow t)) in (types a) /\ (downarrow t) by Th1;
hence a ast t in types a by XBOOLE_0:def 4; :: thesis: verum