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What are the methods of mathematical proof, such as reduction to absurdity and comprehensive analysis, and what else? What are synthesis method and analysis method? Can you give me an example?
Synthesis method and analysis method are common methods in plane geometry.

Deduction from conditional net conclusion and deduction from conditional net conclusion.

Each branch has a different proof method.

For example, infinite descent parity analysis is often used in number theory.

Triangulation is also used in geometry.

The famous inequality method is used to prove inequality and maximum.

The more basic method is direct proof or disproof.