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- Mammoth structures blur the key points.

Example. Compatibility conditions [Spi, vol. 1, p.117, l.1-l.4]. Spivak would help readers gain insight if he had added the following statement: The determinant is smooth across the fibers.

- [Mun00, p.495, l.13-l.14] provides the key point of constructing a
covering space that satisfies the condition given in [Mun00, p.495, Theorem 82.1].

- In order to highlight the key points of a long proof, we should use the following
method rather than repeat the key points many times.
- Find an appropriate example so that the key points may emerge vividly and the reasoning may flow smoothly [Cod, p.139, (1.3)].
- Before we start a proof, we must initiate a plan whose strategy should correspond to that of the above example [Cod, p.152, l.-15-l.-6].
- Pinpoint the exact locations where the key points appear in the proof.

- The key point is often buried in the sea of means that are necessary to make the
formulation precise and accurate.

The key point of [Har, p. 12, Theorem 3.1] is [Har, p.11, Corollary 2.1]. However, in order to use rigorous language to articulate the key idea, we must use two pages [Har, pp.13-14] as means that are necessary to make the formulation precise and accurate.

- We focus on the key point by neglecting trivial matters.

In [Zyg, vol.1, p.61, l.-19], Zygmund says, "The curves y = S_{n}(x) … condense to the interval 0£y£G(p) of the y-axis." What does "condense" mean? Does it mean that these curves stay inside the interval or does it mean that they stay inside a neighborhood of the interval? If we read the general definition of Gibbs' phenomenon [Zyg, vol.1, p.61, l.-12], all we are concerned with is that S_{n}(p/n) ® G(p). Thus, we may neglect the meaning of the word "condense". -
In order to prove [Zyg, vol.1, p.78, Theorem 1.26], we first prove that if n/k
is bounded, then s
_{n, k}® s as s_{n}® s [Zyg, vol.1, p.80, l.4]. Next, we let k = [ne]+1 [Zyg, vol.1, p.80, l.15]. We may ignore the rest of material contained in [Zyg, vol.1, p.79, l.1-p.80, l.11].

- It is difficult to study Tauber's theorem for the first time using the
material in [Ru3, pp.208-209,
§9.1] or that in [Zyg, vol.1, p.81, Theorem
1.36] because the authors add many complications. In essence, Tauber's theorem is simply the converse of
Abel's theorem for certain classes of series [Sak, p.132, l.-12].

- In order to stress the key point, we must hit the nail on the head.

In order to clarify the point, [Wat1, p.365, l.-14-l.-10] hits the nail on the head, while [Guo, p.353, l.-5-l.-4] only touches the point lightly. - The key point to facilitating a theorem's application

For applications, [Perr, p.239, Satz 10] is more convenient than [Perr, p.237, Satz 9]. The key point to this transformation is the inequality given in [Perr, p.238, l.-11].