# You’re weak lemma, you lack importance!

In mathematics we often hear that something is a **Corollary**, something else is a **Lemma**, that is a **Proposition**, there is a **Theorem** … We also have terms such as **Rules**, **Laws**, **Properties**, but these seem to have their general meanings for clearer results, for example: **Sum of Derivatives Rule** … is a result related to the sum operation with derivatives; **Law of Sines** … is a result that determines for any triangle, the relation of the sine of an angle is always proportional to the measure of the opposite side to that angle; **Distributive property of multiplication** … is a result that guarantees a(b + c) = ab + cd.

So, let’s focus on the 4 terms with the most obscure meanings (**Corollary**, **Lemma**, **Proposition**, **Theorem**), what are these names anyway? Let’s start by looking at the dictionary.

**Corollary**: proposition that derives, in a deductive chain, from a preceding assertion, producing an increase of knowledge through the explanation of aspects that, in the previous statement, remained latent or obscure;

**Lemma**: preliminary proposal whose prior demonstration is necessary to demonstrate the main thesis that is intended to be established;

**Proposition**: statement translatable into mathematical symbols, subject to multiple truth values (true, false, indeterminate, etc.) and reducible to two basic elements (the subject and the predicate);

**Theorem**: proposition that can be demonstrated through a logical process.

You may have noticed what they look like. And in fact there is a reason for that, these terms are all **Tautologies**. We will consult in the dictionary what a **Tautology** is.

**Tautology**: analytical proposition that always remains true, since the attribute is a repetition of the subject.

In fact, **Corollaries**, **Lemmas**, **Propositions** and **Theorems**, are all **Tautologies**, that is, sentences whose veracity has been proved in a deductive way. Their differences revolve more around the uses, in this case, the most important results are called **Theorems**. The less important results, but necessary beforehand to prove a theorem, are called **Lemmas**. The results obtained directly from a previous result (either a **Lemma** or a **Theorem**), are called **Corollaries**. And **Propositions** are generally used to describe results of little importance, which are not direct consequences and are not used to demonstrate a **Theorem**. I drafted these ideas to facilitate understanding.

However, this is more a convention than a rule. To say that a result is a **Theorem**, **Corollary**, **Proposition**, **Lemma**, **Rule**, **Property**, **Law** … is to say that this result was demonstrated in a deductive way based on an adopted axiomatic system. In terms of **tautologies**, they are all the same. But in terms of social prestige, **theorems** are superior! And so I justify the cover image of this post, which makes reference to a meme about the anime Naruto, in which Itachi, Sasuke’s brother, defeats and scolds him, saying that he is weak because he lacks hatred! In this case, a **theorem** is rebuking a **lemma**, saying that it is weak, because it lacks importance …

Image adapted of Please Don’t sell My Artwork AS IS from Pixabay