1 1. Remember: you are writing for an expert. Cross out all that is trivial or routine. 2 3 2. Avoid repetition: do not repeat the assumptions of a theorem at the beginning of its proof, or a complicated conclusion at the end of the proof. Do not repeat the assumptionos of a previous theorem in the statement of a next one (instand, write e.g."Under the hypotheses of Theorem 1 with f replaced by g,....."). Do not repeat the same formula -- use a label instead. 4 5 3. Check all formulas: is each of them necessary?
General rules
We denote by $\mathbb{R}$ the set of all real numbers.
We have the following lemma.
The following lemma will be useful.
...... the following inequality is satisfied：
Phrases you can cross out
We denote by $\mathbb{R}$ the set of all real numbers.
We have the following lemma.
The following lemma will be useful.
...... the following inequality is satisfied：
1 Let $\varepsilon$ be an arbitrary but fixed positive number $\Rrightarrow$ Fix $\varepsilon>0$ 2 3 4 5 Let us fix arbitrarily $x\in X$ $\Rrightarrow$ Fix $x\in X$ 6 7 8 9 Let us first observe that $\Rrightarrow$ First observe that 10 11 12 13 We will first compute $\Rrightarrow$ We first compute 14 15 16 17 Hence we have $x=1$ $\Rrightarrow$ Hence $x=1$ 18 19 20 21 Hence it follows that $x=1$ $\Rrightarrow$ Hence $x=1$ 22 23 24 25 Taking into account (4) $\Rrightarrow$ By (4) 26 27 28 29 By virtue of (4) $\Rrightarrow$ By (4) 30 31 32 33 By relation (4) $\Rrightarrow$ By (4) 34 35 36 37 In the interval $[0,1]$ $\Rrightarrow$ in $[0,1]$ 38 39 40 41 There exists a function $f\in C(X)$ $\Rrightarrow$ There exists $f\in C(X)$ 42 43 44 45 For every point $p\in M$ $\Rrightarrow$ For every $p\in M$ 46 47 48 49 It is defined by the formula $F(x)=......$ $\Rrightarrow$ It is defined by $F(x)=......$ 50 51 52 53 Theorem 2 and Theorem 5 $\Rrightarrow$ Theorems 2 and 5 54 55 56 57 This follows from (4),(5),(6) and (7) $\Rrightarrow$ This follows from (4)-(7) 58 59 60 61 For details see [3],[4] and [5] $\Rrightarrow$ For details see [3]-[5] 62 63 64 65 The derivative with respect to $t$ $\Rrightarrow$ The $t-$ derivative 66 67 68 69 A function of class $C^2$ $\Rrightarrow$ A $C^2$ function 70 71 72 73 For arbitrary $x$ $\Rrightarrow$ For all $x$ (For every $x$) 74 75 76 77 In the case $n=5$ $\Rrightarrow$ For $n=5$ 78 79 80 81 This leads to a constradiction with the maximality of $f$ $\Rrightarrow$ .....,contrary to the maximality of $f$ 82 83 84 85 Applying Lemma 1 we conclude that $\Rrightarrow$ Lemma 1 shows that ......, which completes the proof $\Rrightarrow$ .......$\Box$
Phrases you can shorten
Let $\varepsilon$ be an arbitrary but fixed positive number $\Rrightarrow$ Fix $\varepsilon>0$
Let us fix arbitrarily $x\in X$ $\Rrightarrow$ Fix $x\in X$
Let us first observe that $\Rrightarrow$ First observe that
We will first compute $\Rrightarrow$ We first compute
Hence we have $x=1$ $\Rrightarrow$ Hence $x=1$
Hence it follows that $x=1$ $\Rrightarrow$ Hence $x=1$
Taking into account (4) $\Rrightarrow$ By (4)
By virtue of (4) $\Rrightarrow$ By (4)
By relation (4) $\Rrightarrow$ By (4)
In the interval $[0,1]$ $\Rrightarrow$ in $[0,1]$
There exists a function $f\in C(X)$ $\Rrightarrow$ There exists $f\in C(X)$
For every point $p\in M$ $\Rrightarrow$ For every $p\in M$
It is defined by the formula $F(x)=......$ $\Rrightarrow$ It is defined by $F(x)=......$
Theorem 2 and Theorem 5 $\Rrightarrow$ Theorems 2 and 5
This follows from (4),(5),(6) and (7) $\Rrightarrow$ This follows from (4)-(7)
For details see [3],[4] and [5] $\Rrightarrow$ For details see [3]-[5]
The derivative with respect to $t$ $\Rrightarrow$ The $t-$ derivative
A function of class $C^2$ $\Rrightarrow$ A $C^2$ function
For arbitrary $x$ $\Rrightarrow$ For all $x$ (For every $x$)
In the case $n=5$ $\Rrightarrow$ For $n=5$
This leads to a constradiction with the maximality of $f$ $\Rrightarrow$ .....,contrary to the maximality of $f$
Applying Lemma 1 we conclude that $\Rrightarrow$ Lemma 1 shows that ......, which completes the proof $\Rrightarrow$ .......$\Box$