How to Shorten the Paper

 . Remember: you are writing for an expert. Cross out all that is trivial or routine. 

 . 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.

 . 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：

 Let $\varepsilon$ be an arbitrary but fixed positive number $\Rrightarrow$ Fix  $\varepsilon>$

 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 $x=$    $\Rrightarrow$  Hence $x=$

 Hence it follows that  $x=$    $\Rrightarrow$  Hence $x=$

 Taking into account ()   $\Rrightarrow$  By ()

 By virtue of ()   $\Rrightarrow$  By ()

 By relation ()   $\Rrightarrow$  By ()

 In the interval $[,]$   $\Rrightarrow$  in $[,]$

 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  and Theorem    $\Rrightarrow$  Theorems  and 

 This follows from (),(),() and ()   $\Rrightarrow$  This follows from ()-()

 For details see  [],[] and []   $\Rrightarrow$  For details see []-[]

 The derivative with respect to $t$   $\Rrightarrow$  The $t-$ derivative

 A function of class $C^$   $\Rrightarrow$  A $C^$ function

 For arbitrary $x$   $\Rrightarrow$  For all $x$ (For every  $x$)

 In the case $n=$   $\Rrightarrow$  For $n=$

 This leads to  a constradiction with the maximality of $f$   $\Rrightarrow$  .....,contrary to the maximality of $f$

 Applying Lemma  we conclude that   $\Rrightarrow$  Lemma  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  $\Rrightarrow$ .......$\Box$