Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$

Suppose $v+W=v^{\mathrm{\prime }}+W$. By (W1) of [L2] we have $v=v+0_V\in v+W$, so $v\in v^{\mathrm{\prime }}+W$, giving $w\in W$ with $v=v^{\mathrm{\prime }}+w$ and hence $v-v^{\mathrm{\prime }}=w\in W$. Conversely suppose $v-v^{\mathrm{\prime }}\in W$. For $w\in W$, $v+w=v^{\mathrm{\prime }}+((v-v^{\mathrm{\prime }})+w)$ and $(v-v^{\mathrm{\prime }})+w\in W$ by (W2), so $v+W\subseteq v^{\mathrm{\prime }}+W$; since $v^{\mathrm{\prime }}-v=(-1)(v-v^{\mathrm{\prime }})\in W$ by (W3), the same argument gives $v^{\mathrm{\prime }}+W\subseteq v+W$. Hence $v+W=v^{\mathrm{\prime }}+W$ exactly when $v-v^{\mathrm{\prime }}\in W$.

Let $v+W=v^{\mathrm{\prime }}+W$ and $u+W=u^{\mathrm{\prime }}+W$. By step 1.1, $v-v^{\mathrm{\prime }}\in W$ and $u-u^{\mathrm{\prime }}\in W$, so $(v+u)-(v^{\mathrm{\prime }}+u^{\mathrm{\prime }})=(v-v^{\mathrm{\prime }})+(u-u^{\mathrm{\prime }})\in W$ by (W2) and $av-a{v}^{\mathrm{\prime }}=a(v-v^{\mathrm{\prime }})\in W$ by (W3). Applying step 1.1 in the converse direction gives $(v+u)+W=(v^{\mathrm{\prime }}+u^{\mathrm{\prime }})+W$ and $(av)+W=(a{v}^{\mathrm{\prime }})+W$, so both quotient operations are independent of representatives.

The vector-space identities in $V\mathrm{/}W$ follow by applying the corresponding identities in $V$ to representatives, with zero coset $W=0_V+W$ and inverse $-(v+W)=(-v)+W$; moreover $\pi (av+bu)=(av+bu)+W=a\pi (v)+b\pi (u)$ by [L1] and [L3], every coset is $\pi (v)$ by definition, and by step 1.1 $\pi (v)=0_V+W$ exactly when $v-0_V=v\in W$, so $\pi$ is linear and surjective with $\ker \pi =W$.