Let $\mathcal{C}$ be a highest-weight category with $\Lambda$ as a interval-finite poset -- I'm using a definition of the highest-weight category given by Cline, Parshall and Scott and it is presented also on the Wikipedia site: Highest-weight category. In the paper of authors given above it was presented the lemma:

Let $\lambda,\mu \in \Lambda.$ Then:

(a) $S(\lambda)$ is the socle of $A(\lambda)$ (it is clear from third axiom),

(b) If either $\text{Ext}^1_{\mathcal{C}}(A(\mu),A(\lambda))$ or $\text{Ext}^1_{\mathcal{C}}(S(\mu),A(\lambda))$ is nonzero, then necessarily $\mu>\lambda.$ If $\text{Ext}^1_{\mathcal{C}}(S(\mu),S(\lambda))\neq 0,$ then $\mu$ and $\lambda$ are strictly comparable.

(c) The filtration $\{F_n(\lambda)\}$ in third axiom can be chosen to satisfy the additional condition: for all $i,j>0$, if $\mu_i<\mu_j$ then $i<j.$

It is written that "clearly, (b) implies (c)." I have to admit that it isn't clearly for me (maybe it is very easy though). I don't have any idea at this moment how to construct a filtration with ordered $\mu_i.$ I have thought a few hours but I didn't solve it. It seems for me that (a) could be a key result, but I'm stuck at this point. I would be grateful for any help, hints etc.