Proof of The Origin of ℝⁿ Has Dot Product Zero with Every Point

The operations and the origin of $\mathbb{R}^{n}$ are the operations of Pointwise Operations on Real-Valued Maps (i) and the constant map $0$ of Pointwise Operations on Real-Valued Maps (ii) with $I=[n]$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv) and Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (v). Hence Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (iv), with $I=[n]$ and $f=x$, gives $0x=0_{n}$. The scalar rule of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (ii), applied with $x$ as the multiplied vector, $t=0$ and $z=x$, then gives

$$0_{n}\cdot x=(0x)\cdot x=0\,(x\cdot x).$$

Finally $0\,(x\cdot x)=0$ by Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (vii) ($x\cdot x$ being a real number by The Dot Product on $\mathbb{R}^n$ (i)), so $0_{n}\cdot x=0$, which is (i).