• How much we know about the related topic which we are going to build new theory on.
  • What information are possibly useful for proving or disproving a new idea.
  • The underlying assumptions of existing theorems, lemmas, corollaries and proposition even definitions, what are the essential factors which have been used to derive theorems, lemmas, corollaries and propositions. 
  • Can we modify the essential factors to get weak or strong generalization. The methods used to prove, what are the advantage and disadvantage ( or usage of the methods).
  • Methods are used based on the information we have to prove or disprove the claim. Every claim , definition… has its meaning, physical meaning or whatever, there is meaning, intuition with what we claimed to be true or false, otherwise, useless.To be able to build a new theory, prove what is to be true based on observation, logical thinking, to grasp the content, methods and meaning of mathematics has its very significance.  
  • To construct or derive from underlying assumption sth as close as to the conditions in the proved theorems, lemma or corollary, then we could apply those theorem, lemma or corollary as tool to prove our claim. 
  • Mathematic proof is a rigorous logic construction based on the proved factors. 
  • Decompose the big information into small piece informations for further analysis. 
  • Prove by contradiction, prove by induction, prove by deduction…..1*10^{-157}