Proportion is Equivalence Relation
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Theorem
Proportion is an equivalence relation.
Proof
- Proportion is Reflexive: $\forall x \in \R: x \propto x$
- Proportion is Symmetric: $\forall x, y \in \R: x \propto y \implies y \propto x$
- Proportion is Transitive: $\forall x, y, z \in \R: x \propto y \land y \propto z \implies x \propto z$
The result follows from the definition of an equivalence relation.
$\blacksquare$