The following question assumes familiarity with the discrete model described in chapter 5 of Steven Roman's "Introduction to the Mathematics of Finance", 2nd edition, Springer 2012. I will not describe the model or the associated notation in this post.
The Law of One Price (p. 132) states that, in the absence of arbitrage opportunity in the market, $$ \mathcal{V}_T(\Phi_1) = \mathcal{V}_T(\Phi_2) \implies \mathcal{V}_k(\Phi_1) = \mathcal{V}_k(\Phi_2) $$ for all times $0 \leq k \leq T$ and for all self-financing trading strategies $\Phi_1$ and $\Phi_2$.
Unfortunately, no proof is provided in the text (in fact, this law is stated as a definition rather than a theorem). Why does this law hold?
Additionally, it is implied by the text following the statement of the Law of One Price, that, if the market has no arbitrage opportunity, then, given an attainable alternative $X$, if a new asset $a^*$ is introduced into the market and is priced in such a way that its payoff at time $t_T$ is $X$ and its pricing is consistent with the Law of One Price, i.e. for every $k \in \{0, 1, \dots, T\}$, $S_{a^*, k} := \mathcal{V}_k(\Phi)$, where $\Phi$ is any self-financing trading strategy such that $\mathcal{V}_T(\Phi) = X$, then the resulting, extended market will still have no arbitrage opportunity.
Why is this so?