Lemmas and theorems are both important concepts in mathematics. The main differences are:
1. The theorem has a higher status and represents an important conclusion or proposition. Theorems usually require rigorous proof and express core knowledge in a certain field.
2. Lemmas are of secondary importance and exist to assist in the proof of theorems . The conclusion expressed by a lemma may be subtle or may be a step in the theorem proving process.
3. The theorem must be proved strictly, while the lemma sometimes does not need proof or only needs simple proof .
4. Theorems are based on postulates, axioms, known theorems, etc., and lemmas are often based on a theorem and are used to derive the proof of the theorem.
5. In articles and papers, theorems should be expressed as "theorem", while lemmas can be expressed as "lemma", "proposition" or "property".
6. A theorem may generate many related lemmas, but a lemma will not generate a theorem .
7. Theorems are the key components of the theoretical framework, and lemmas play a supporting role. A good theorem should have broad applicability. In summary, lemmas and theorems complement each other and jointly promote the development of mathematical theoretical systems. Lemma provides support for theorem, and theorem can express important and profound conclusions.