Existence of Transformations whose Commutator equals Identity

Theorem
Let $\map {C^\infty} \R$ be the space of smooth real functions.

Let $\circ$ denote the composition of mappings.

Let $A : \map {C^\infty} \R \to \map {C^\infty} \R$ be the mapping such that:


 * $\forall \phi \in \map {C^\infty} \R : \forall x \in \R : \map {\paren{ A \circ \phi } } x := \map {\dfrac {d \phi} {d x} } x$

Let $B : \map {C^\infty} \R \to \map {C^\infty} \R$ be the mapping such that


 * $\forall \phi \in \map {C^\infty} \R : \forall x \in \R : \map {\paren{ B \circ \phi } } x := x \map \phi x$

Let $I : \map {C^\infty} \R \to \map {C^\infty} \R$ be the identity such that:


 * $\forall \phi \in \map {C^\infty} \R : I \circ \phi = \phi$

Then:


 * $\forall \Psi \in \map {C^\infty} \R : \paren {A \circ B - B \circ A} \circ \Psi = I \circ \Psi$

or with slight abuse of notation:


 * $A \circ B - B \circ A = I$

Proof
Let $\Psi \in \map {C^\infty} \R$.

Then: