). It follows that s (cid:7)= s
(cid:10)
(cid:10) ∈ f2(T ) \ f2(T
(cid:10)
). Suppose that t = t
(cid:10)(cid:10)
(cid:10) ∈ f1(s), t

(cid:10) such that u ∈ f2(t
). However, it must then hold that also u

) = f2(T
(cid:10)
) and u
(cid:10)(cid:10)
(cid:10) ∈ f2(t
(cid:10). We now have t, t

(cid:10) and T ∩ T
). There are, thus, t, t
(cid:10). Then u

). There is also some t
(cid:10)(cid:10)

(cid:10)(cid:10) ∈ f2(u) and t

/∈ f1(s), t, t

(cid:10) ∈ f3(s

(cid:10)(cid:10) ∈ T

(cid:10)(cid:10)

(cid:10)

(cid:10)

The following condition is an immediate consequence.

Corollary 60. A composition  f1◦ f2 of two transformation functions  f1 and  f2 is a transformation function if either  f1 is M↑ or  f2 is 
M↓.

While Theorem 59 is a suﬃcient condition, it is not a necessary one, as the following example shows.
A-framework-for-analysing-state-abstraction-metho_2022_Artificial-Intelligen