Automata: Proof

Question

Here is the problem:

Consider a NFA, M = (K, Σ, Δ, s, F) with (p, a, q) ∈ Δ. Prove that (pʹ, aw) ⊢∗ (qʹ, w) for any w ∈ Σ∗, q′ ∈ E(q) and p′ with p ∈ E(p′).

Thanks in advance.

Answer 1

Since $p'\in E(p)$ and $q'\in E(q)$, $(p',aw)\vdash(p,aw)\vdash(q,w)\vdash(q',w)$, using the transitions $(p',\epsilon,p),(p,a,q)$, and $(q,\epsilon,q')$ in that order.