---
tag: 0NTF6DMVAP
kind: concept
name: Direct proof
msc: [00A35]
uses: [0B94TKFQJX, 0ZNX6DQ3EJ]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_proof
    locator: lead section
same_as:
  wikidata: Q3240280
  wikipedia: Direct proof
---

## Summary

A direct proof of an [[implication|0ZNX6DQ3EJ]] $p \to q$ assumes $p$ and then gets $q$ by a chain of valid steps.
The steps use axioms, definitions and theorems that are already proved.

## Remarks

Example: if $n$ is odd, then $n^2$ is odd. Write $n = 2k + 1$. Then $n^2 = 2(2k^2 + 2k) + 1$, which is odd.
When a direct proof is hard, a [[proof by contrapositive|0BB5TPGQ1N]] or a [[proof by contradiction|0J0XBKG64S]] can help.

## Explanations

### L1 human

In a direct proof you go straight from the hypothesis to the conclusion.
You write down what the hypothesis means, then you compute or reason until you reach the conclusion.