1. Purpose and goal: define the purpose and goal of the proposition, and determine the problem to be solved or the conclusion to be proved.
2. Preconditions: Determine the preconditions of a proposition, that is, known facts, theorems or axioms. These conditions are the basis of derivation and proof.
3. Logical relationship: consider the logical relationship between the elements in the proposition, including equality, inequality, necessary and sufficient conditions, etc.
4. Symbols and representations: Choose appropriate symbols and representations to describe the proposition, making it clear, concise and easy to understand.
5. Structure and form: determine the structure of a proposition, such as whether the proposition is a theorem, lemma or inference, and the form of the proposition, such as whether the proposition is an identity, an existential proposition or a full-name proposition.
6. Reasoning proof: the correctness of the proposition is proved by using logical reasoning and mathematical methods. This may involve using existing theorems, axioms or properties, or developing new proof methods.
7. Generality and particularity: consider the generality and particularity of the proposition, that is, whether the proposition is applicable to all situations or only to specific situations.
8. Popularization and application: think about how to popularize the proposition to a wider range of situations and explore its application in practical problems.
9. Integrity and consistency: ensure the integrity and consistency of the proposition, that is, the proposition does not contain contradictory or self-contradictory elements.
10. falsifiability: Consider the falsifiability of the proposition, that is, whether there are counterexamples and whether the proposition is wrong.
To sum up, to construct a mathematical proposition, we need to comprehensively consider the goal, premise, logical relationship, symbol and representation, structure and form, reasoning and proof, generality and particularity, popularization and application, completeness and consistency, and falsifiability.