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Please rate and comment. I want to improve; constructive criticism is highly appreciated.

Please take style into account as well.

The following proof is solely based on vector space axioms.

Axiom names are *italicised*.

They are defined in Wikipedia (vector space).

- Distance from point $(1,1,1,1)$ to the subspace of $R^4$
- Image of dual map is annihilator of kernel
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We prove the uniqueness of an identity element of addition of a vector space.

Let $V$ be a vector space.

It remains to prove that an identity element of addition of $V$ is unique.

We define a predicate $P$ over $V$ as follows.

For every $v \in V$,

$$P(v) \quad := \quad \big(\, u + v = u \; \text{ for all } u \in V \,\big).$$

By *Identity element of addition*,

there exists an element of $V$ for which $P$ holds.

Let $0 \in V$ such that $P(0 )$ holds;

let $0′ \in V$ such that $P(0′)$ holds.

It remains to prove that $0 = 0’$.

\begin{align*}

0 & = 0 + 0′ && \text{since } P(0′) \text{ holds.} \\

& = 0′ + 0 && \text{by }\textit{Commutativity of addition.}\ \\

& = 0′ && \text{since } P(0) \text{ holds.}

\end{align*}

QED

P.S.: This is intended to improve my previous, roundabout proof.

Of course, I could only incorporate a small subset of the helpful advice given there.

- Prove projection is self adjoint if and only if kernel and image are orthogonal complements
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- What is (fundamentally) a coordinate system ?

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