Phrases or clauses frequently used in mathematics

常见的涉及运算的短语或句子 

1. Differentiate both sides of the equation and we get...

   Integrate  both sides of the equation and we get...

2. Differentiating both sides of the equation , we get...

Integrating both sides of the equation , we get...

3.            Add (A) to (B) and we have…   

(其中(A) 、(B)为某些表达式，如不等式、等式、方程等，下同)

4．Subtract  (B)  from  (A)  and  we have …

5. Multiplying each term of the equation by …, we obtain…

6. Dividing the equation through by ..., we have ...

7.   (A) and (B)  together   give  …             

(A) and (B)  together   yield …     

(A) and (B)  together   imply …     

8.  Comparing (A) with (B), it is easy to see that …

9.  Substituting (A) into (B), we obtain …

10. Eliminating (the parameter) t from (A) and (B),we have …

11. By introducing a new variable …, we can then rewrite (A) as follows

    By introducing a new variable …, we can then rewrite (A) in the following form

12. By a simple calculation, we obtain from (A) …

定理证明过程中常见的短语和句子 

1.下面的句型可用来表达“根据什么即可得到什么”的意思

According to definition , it follows … 

According to hypothesis , it follows … 

According to asssumptions, it follows … 

According to theorem(N), it follows … 

According to lemma (A) , it follows … 

According to corollary (B) , it follows … 

According to the remark , it follows … 

According to the fact that … , it follows … 

(可以把上面的“according to ”换成“ by” )

Since …, it follows …

2. 如果一个论断可以通过一些简单运算或简单推理而获得，由于这些运算或推理比较简单，读者可以自行推算，因而只需直接写出论断来，这时可用下面句型：

(1)  It is easy to see that …

     It is easy to show that …

     It is easy to prove that …

     It is easy to verify that …

 It is easy to check that …

(2)  It can easily be seen that …

It can easily be shown that …

It can easily be proved that …

It can easily be verified that …

It can easily be checked that … 

3.如果所要提及的结论比较显浅，或是众所周知，无需作进一步的证明，这时可用下面句型：

 (1)   It is clear that …

It is obvious that …

It is evident that …

It is well-known that …

(2)   Clearly, …

      Obviously, …

      Evidently,…

4.为了证明一个定理有时需要引进辅助函数，这时可用下面句型：

 Let us first define the function…

Let us introduce a new function…

   Let us consider the function…

Let us first  investigate the function…

Let …

Set…

Define…

Put…

Consider…

5. 在一个定理中，有几个结论需要证明，其中有些结论比较明显，可不用证明，仅需证明余下结论即可，这时可用下面句型：

 Since (A) and (B) are obvious, we need only prove (C).

Since (A) and (B) are trivial, we need only prove (C).

 Since (A) and (B) are trivial, it suffices to prove (C)

6. 为了证明一个定理，有时我们并不是直接去证明，而是证明一个新的论断，一旦新的论断得到证明，已给定理不难由此而得证，这时可用下面句型：

 以下各句用于新的论断被证明之前：

The theorem will be proved if we can show…

The result will be proved if we can show…

The theorem will be proved by showing that…

If we can prove…then the theorem follows immediately.

以下各句用于新的论断被证明之后：

The theorem is now a direct consequence of what we have proved.

The theorem follows immediately from what we have proved.

The theorem is now evident from what we have proved.

It is evident to see that the theorem holds.

7.在证明过程中，有时要用到一些早已学过的知识或技巧，这时可用下面句子，以提醒读者：

 Recall that…

Notice that…

Note that…

Observe that…

In order to prove the theorem, we need the knowledge of …

In order to obtain the following equation, we need…

8. 如果需要证明的定理的假设条件是一般条件，但是，只要定理在特殊条件下成立，就不难推出定理在一般条件下也成立，这时仅需要在特殊情况下去证明定理就够了，为此可用下面句型：

   Without loss of generality, we may consider…

   Without loss of generality, we may assume…

   Without loss of generality, we may prove the theorem in the case…

   It suffices to prove the theorem in the case…

   We need only consider the case…

   For simplicity, we may take…

9. 如果待证的论断可用以前用过的相似的方法或步骤进行证明，则可用下面句型：

   This theorem can be proved in the same way as shown before.

   This statement can be proved in a similar way as shown before.

   This theorem can be proved by the same method as employed in the last section.

   This theorem can be completed by the method analogous to that used above.

   Using the same argument as in the proof of theorem N, we can easily carry out the proof of this theorem.

   We now proceed as in the proof of theorem N.

   We shall adopt the same procedure as in the proof of theorem N.

10. 如果我们用的是反证法，则其开头及结尾可用下面句型：

    If the statement(or assertion, conclusion) were false(or not true, not right) then…

    If the assertion would not hold, then…

    This is contrary to…

    This contradicts the fact that…

    This leads to a contradiction.

11. 表示定理已证毕或者把前面所证的总结为一结论

    We have thus proved the theorem.

    This completes the proof.

    The proof of the theorem is now completed.

    It is now obvious that the theorem holds.

    Thus we have derived that …

    Consequently, we infer that…

    Thus we conclude that…

    Thus we are led to the conclusion that …

    Thus we arrive at the conclusion that …

    Thus we can summarize what we have proved as the following theorem.

12. 其它

     There exist(s)…such that…

     We claim…in fact…

     We are now in a position to…

     If otherwise…

     Provided that…