- Origins of theorems
- A general criterion (see [Po3, p.495, D)] and its application [Po3, p.495, Theorem 111]).
- Characterizing the basics (Making the hypothesis as weak as possible) [Po3, p.471, l.!2-p.472, l.3].

- A simple, elementary and commonly used theorem should be introduced before complicated and advanced theorems. Courant introduced the theorem on change of order of differentiation in the beginning of his book [Cou2, vol.2, pp.36-37] while Rudin places this elementary and commonly used theorem [Ru1, Theorem 9.34] after the difficult theorems [Ru1, Theorems 9.17, 9.18, 9.19, 9.20 & 9.21]. Rudin's disordering might cause the reader to mistakenly think that knowledge of these difficult theorems is a prerequisite for understanding the simple theorem [Ru1, Theorem 9.34].

- The statement of a theorem should be simple and able to capture the theorem's
essential features.

Example. The statement of Jordan's lemma in [Gon, pp.680-681, Lemma 9.2] is more concise and inclusive than the version in [Ant, p.239, Lemma 6.2.1]. Since the residue theorem is ultimately rooted on the contour's parameterization, the use of polar coordinates is a natural way to describe an arc. Obviously, Antimirov prefers to cut an arc from a circle horizontally. However, the type of cut is irrelevant to this lemma. Antimirov's preference only complicates the theorem's description and proof.

- A theorem should not be formulated for a special purpose. It should be formulated for wide usage. Both [Kre.
p.238, l.-14] and [Kre, p.238, l.-9]
use [Lau, p.67, Theorem 6.2.1], a better formulation of [Kre, p.205, Theorem
66.1]. This is because the statement of [Kre, p.205, Theorem 66.1] cannot be
directly used to prove [Kre. p.238, l.-14] and [Kre,
p.238, l.-9]. Only the idea of the proof of [Kre,
p.205, Theorem 66.1] is good for proving these two cases.

- In terms of sufficient conditions, [Sne, p.21, Theorem 5] is a refinement of [Cou2, vol. 2, p.104, the
fundamental theorem] because the sufficient condition of the former theorem is
weaker than that of the latter theorem. If a screw in your eyeglasses loosens,
a precision screwdriver would be needed to tighten it. An ordinary screwdriver would be too rough for
the job. Even though the latter theorem can handle most cases, the former
theorem can handle more delicate cases [Sne, p.27, Example 7; p.29, Example 10;
p.30, Example 11]. A refinement does not necessarily mean
it is more difficult. From the viewpoint of elementary calculus, the former
theorem is certainly difficult. However, from the viewpoint of vector analysis,
the structure of the former theorem is very simple. Thus, for a refinement, we have to view it
from a different perspective and handle it with another set of tools.

Remark. When a theorem is written in the "if and only if" form, it does not mean that one has completely investigated the theorem even though one has traveled both ways and has not missed any stations in between. One must make the sufficient condition as weak as possible and make the necessary condition as strong as possible [Cou2, vol. 2, p.84, (48)].

- In order to make a theorem [Sag, p.38, Theorem 1.8.2] meaningful, we must
use a concrete example to justify each hypothesis of the theorem.

Example 1. (The theory of extreme values in differential calculus) [Sag, p.42, l.10-l.11].

Example 2. (Calculus of variations) [Sag, p.34, l.-13-l.-1; p.37, l.9-l.-7].

### A theorem's formulation should focus on the essentials rather trivialities.

Example. (The inverse function theorem: (f^{-1})'(y)=[f'(f^{-1}(y))]^{-1}) Modern mathematicians strive to distinguish themselves. They make a fuss over the trivial aspects of a theorem but fail to mention anything about its essential content. The formulation of the inverse function theorem given in [Spi1, p.35, Theorem 2-11] focuses on the essential content, while that given in [Ru1, p.193, Theorem 9.17], [Tay, p.256, Theorem I; Cou2, vol.2, p.261, l.12-l.-1], [O'N, p.39, Theorem 7.10] and [Gun, p.17, Theorem 7] does not. Applications of the inverse function theorem use the theorem's various versions: [Sag, p.109, (A2.14.6)] uses the version (fÎC^{n}Þ f^{-1}ÎC^{n}); [O'N, p.39, Theorem 7.10] uses the version (fÎC^{¥}Þ f^{-1}ÎC^{¥}); [Gun, p.17, Theorem 7] uses the version (If f is analytic, then f^{-1}is analytic). However, these versions of the inverse function theorem can be easily derived from the formula given above.- The order of priorities for a theorem's formulation: The essentials ®
generalization ® various ways the theorem applies to practical
problems.

The equality given in [Sag, p.120, l.-9] could be directly proved by using [Kap, p.267, Theorem], but the formulation of [Kap, p.267, Theorem] is too complicated. It is difficult to recognize the essential meaning of [Kap, p.267, Theorem]. In contrast, the formulation of [Wid, p.353, Theorem 11] focuses on the essentials, so it is simple and easy to remember. In [Sag, p.120, l.-11-l.-10], Sagan wisely uses it with the help of the chain rule.

- We find a theorem's partial converse by reversing the theorem's proof step by step.
If a reasoning step is one-way only, then we try to impose certain conditions in
order to make it two-way.

Example. The converse of [Har, p.73, Theorem 11.1] is false [Har, p.74, l.-18]. When we reverse the steps in the proof of [Har, p.73, Theorem 11.1], we must apply the factorization theorem for polynomials to matrices. [Har, p.73, Corollary 11.1] gives the general case for matrix factorization. [Har, p.75, Lemma 11.2] gives the special case when

det X(0)=0.

- The Mittag-Leffler theorem can be formulated into versions with different emphases.
The version we use for applications should be in tune with the circumstance. If
we want to expand a meromorphic function in rational fractions from scratch, we
use [Guo, p.17, l.-13-p.18, l.4]. If the poles and
their corresponding principal
parts of the meromorphic function are given, we use [Gon1, p.286, Theorem 4.3] [1].
The same discussion applies to [Guo, p.25, Theorem 1] and [Gon1, p.202, Theorem
3.16] [2].

- [Wat1, p.447, l.17-l.11] says that readers may obtain results (I) and (II) using the methods given in previous sections.
First, Ã(z) is a
series, while s(z) is an infinite product. It
requires [Ru2, p.323, Theorem 15.6] to explain why these two different entities
are related to the same topic. Second, The
derivation of uniform and absolute convergence requires an articulation of the
conditions of the circumstance (formulation of the exact hypotheses) and a rigorous
proof. To help us execute the idea we need a theorem [Ru2, p.324, Theorem 5.9] instead of a method. For problems of infinite products in complex analysis, it
is sufficient to use the above theorems. For problems of infinite products in
the theory of elliptic functions, we need an extra fact: [Gon1, p.292, l.3-l.4].

- If Theorem B is used to prove Theorem A, we should choose the strongest form
of theorem that satisfies the hypothesis of Theorem A for the following reasons:
- We can fully use the available resources.
- We try to obtain the most effective method to prove Theorem A.

- One should always ask why a theorem's conclusion requires the conditions
in the hypothesis. This approach may help us eliminate superfluous
assumptions. Sometimes only when we reach the end of the proof may we discover
the true reason. Sometimes we have to look for our answer in the proof of a
quoted theorem.

Example. One may wonder why [Zyg, vol.1, p.105, Theorem 8.1] requires the assumption a£1. In order to prove the inequality given in [Zyg, vol.1, p.106, l.11], we need use [Zyg, vol.1, p.94, (5.5)(ii)] whose proof requires a£1 [Zyg, vol.1, p.94, l.13, l.16 & l.-4].

- Theorems vs. their applications [Perr, p.291, l.16-l.22]
- Links {1,
2,
3}.