Right Distributive Law for Natural Numbers

Theorem
The operation of multiplication is right distributive over addition on the set of natural numbers $\N_{> 0}$:
 * $\forall x, y, n \in \N_{> 0}: \paren {x + y} \times n = \paren {x \times n} + \paren {y \times n}$

Proof
Using the axiomatization:

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $\forall a, b \in \N_{> 0}: \paren {a + b} \times n = \paren {a \times n} + \paren {b \times n}$

Basis for the Induction
$\map P 1$ is the case:

and so $\map P 1$ holds.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * $\forall a, b \in \N_{> 0}: \paren {a + b} \times k = \paren {a \times k} + \paren {b \times k}$

Then we need to show:
 * $\forall a, b \in \N_{> 0}: \paren {a + b} \times \paren {k + 1} = \paren {a \times \paren {k + 1} } + \paren {b \times \paren {k + 1} }$

Induction Step
This is our induction step:

The result follows by the Principle of Mathematical Induction.

Also see

 * Left Distributive Law for Natural Numbers