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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.

For the theorem that the product of two even numbers is even, the proof by contradiction assumes the opposite: that the product of two even numbers is odd. The speaker justifies this by stating that if a number is not even, it must be odd.

Conditions: The theorem to be proved is: the product of two even numbers is also even.; The method is proof by contradiction.; The parity dichotomy (not even implies odd) is used as an assumed fact.

For the theorem that the product of two even numbers is even, the proof by contradiction assumes the opposite: that the product of two even numbers is odd. The speaker justifies this by stating that if a number is not even, it must be odd.

Conditions: The theorem to be proved is: the product of two even numbers is also even.; The method is proof by contradiction.; The parity dichotomy (not even implies odd) is used as an assumed fact.

The presenter switches from numerical examples to an algebraic proof because checking specific cases like $6\times8=48$ and $2\times4=8$ only verifies the statement for those particular pairs. To prove the universal statement that the product of *any* two even numbers is even, a general algebraic argument is required to cover all possible even numbers.

Conditions: The goal is to prove a universal statement about all even numbers.; Numerical examples are used only to build intuition.; Algebra is necessary for a general proof.

The presenter switches from numerical examples to an algebraic proof because checking specific cases like $6\times8=48$ and $2\times4=8$ only verifies the statement for those particular pairs. To prove the universal statement that the product of *any* two even numbers is even, a general algebraic argument is required to cover all possible even numbers.

Conditions: The goal is to prove a universal statement about all even numbers.; Numerical examples are used only to build intuition.; Algebra is necessary for a general proof.