Bridgeland stability conditions: The heart satisfies the Harder-Narasimhan property Given a stability condition $(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$. Take $\mathcal{A}=\mathcal{P}((0,1])$. Then $\mathcal{A}$ is the heart of a bounded t-structure on $\mathcal{D}$.

Why can the Grothendieck group $K(\mathcal{A})$ be identified with $K(\mathcal{D})$?
How does this help prove that any object in $\mathcal{A}$ has a Harder-Narasimham filtration by $\left. Z\right|_\mathcal{A}$-stable objects?

I'm refering to the second last paragraph of page 12 in the paper:
http://arxiv.org/pdf/math/0611510v1.pdf
Thank you very much.
 A: I have only recently worked through Bridgeland's paper, so I apologize if I make a mistake, but I think this is Proposition 5.3 from page 15 of Bridgeland's previous paper Stability Conditions on Triangulated Categories, which I will refer to below.

Proposition 5.3. To give a stability condition on a triangulated category $\mathcal{D}$ is equivalent to giving a bounded $t$-structure on $\mathcal{D}$ and a stability function on its heart with the Harder-Narasimhan property.

The reason $K(\mathcal{A}) = K(\mathcal{D})$ comes from Lemma 3.2:

Lemma 3.2. Let $\mathcal{A} \subset \mathcal{D}$ be a full additive subcategory of a triangulated category $\mathcal{D}$. Then $\mathcal{A}$ is the heart of a bounded $t$-structure $\mathcal{F} \subset \mathcal{D}$ if and only if the following two conditions hold:
(a) if $k_1 > k_2$ are integers and $A$ and $B$ are objects of $\mathcal{A}$ then $\text{Hom}_{\mathcal{D}}(A[k_1],B[k_2]) = 0$.
(b) for every nonzero object $E \in \mathcal{D}$ there is a finite sequence of integers $k_1 > \cdots > k_n$ and a collection of triangles $E_0 \to E_1 \to A_1 \to, \ldots, E_{n-1}\to E_n \to A_n\to$ (with $E_0 = 0, E_n = E$) with $A_j \in \mathcal{A}[k_j]$ for all $j$.

So every object $E \in \mathcal{D}$ is built out of objects in $\mathcal{A}$ by (b), and in $K(\mathcal{D})$ one has
$$[E] = [E_n] = [E_{n-1}] + [A_{n}] = \cdots = \sum_{i=1}^n [A_i] \in K(\mathcal{A}).$$
The filtration from definition 3.3(c) is a Harder-Narasimhan filtration. The objects $E_i$ are clearly isomorphic to ones in $\mathcal{A}$, since $E_{n-1}$ is the kernel of $E \to A_n$ both of which are in $\mathcal{A}$, and $E_i/E_{i-1} \cong A_i \in \mathcal{P}(\phi_i)$ are semistable.
