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In the proof, arbitrary even numbers are represented algebraically as twice another natural number. Specifically, the first even number $x$ is written as $x=2c$ and the second even number $y$ is written as $y=2p$, where $c$ and $p$ are natural numbers ($c,p \in \mathbb{N}$).

Conditions: The numbers $x$ and $y$ are assumed to be even.; $c$ and $p$ are natural numbers ($\mathbb{N}$).; The definition of an even number is applied: any even number is twice another number.

In the proof, arbitrary even numbers are represented algebraically as twice another natural number. Specifically, the first even number $x$ is written as $x=2c$ and the second even number $y$ is written as $y=2p$, where $c$ and $p$ are natural numbers ($c,p \in \mathbb{N}$).

Conditions: The numbers $x$ and $y$ are assumed to be even.; $c$ and $p$ are natural numbers ($\mathbb{N}$).; The definition of an even number is applied: any even number is twice another number.

The product of two even numbers is rewritten by substituting their algebraic representations ($x=2c$ and $y=2p$) into the product expression $xy$. This gives $(2c)(2p)$, which simplifies to $4cp$.

Conditions: $x=2c$ and $y=2p$ with $c,p \in \mathbb{N}$.; Basic arithmetic properties of multiplication (associativity and commutativity) are used.; The goal is to express the product in the form $2k$ to show it is even.

The product of two even numbers is rewritten by substituting their algebraic representations ($x=2c$ and $y=2p$) into the product expression $xy$. This gives $(2c)(2p)$, which simplifies to $4cp$.

Conditions: $x=2c$ and $y=2p$ with $c,p \in \mathbb{N}$.; Basic arithmetic properties of multiplication (associativity and commutativity) are used.; The goal is to express the product in the form $2k$ to show it is even.