Vector space/Linear subspace/Translation/Equivalence relation/Fact/Proof

Proof

We look at the conditions of an equivalence relationn The reflexivity follows from $v-v=0\in U$, the symmetry follows from $w-v=-(v-w)\in U$. The transitivity follows like this: from $u-v\in U$ and $v-w\in U$, we infer $u-w=(u-v)+(v-w)\in U$.

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