Proportion is Transitive
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Theorem
Proportion is a transitive relation.
That is:
- $\forall x,y,z \in \R: x \propto y \land y \propto z \implies x \propto z$
Proof
Let $x, y, z$ be arbitrary.
Let $x$ be proportional to $y$ and $y$ to $z$:
- $x \propto y \land y \propto z$
Then by definition:
- $\exists j, k \ne 0: x = j \times y \land y = k \times z$
Substituting $k \times z$ for $y$:
- $x = \left({j \times k}\right) \times z$
so $j \times k$ is the desired constant of proportion.
The result follows from the definition of transitivity and proportion.
$\blacksquare$