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1、Phrases or clauses frequently used in mathematics （转载）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 w
2、e have (其中 (A) 、(B)为某些表达式，如不等式、等式、方程等，下同) 4Subtract (B) from (A) and we have ,5. Multiplying each term of the equation by , we obtain6. 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 ea
3、sy 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 followsBy introducing a new variable , we can then rewrite (A) in the following form12. By a simple calculation, we obtain
4、from (A) 定理证明过程中常见的短语和句子名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 1 页，共 7 页 - - - - - - - - - 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 f
5、ollows 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
6、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 名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - -
7、- - - 第 2 页，共 7 页 - - - - - - - - - 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 functionLet us introdu
8、ce a new functionLet us consider the functionLet us first investigate the functionLet Set Define Put Consider 名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 3 页，共 7 页 - - - - - - - - - 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 obv
9、ious, we need only prove ?. Since (A) and (B) are trivial, we need only prove ?. Since (A) and (B) are trivial, it suffices to prove ? 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 sho
10、w 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 evid
11、ent 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 thatNotice thatNote that名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 4 页，共 7 页 - - - - - - - - - Observe thatIn order to prove the theorem, we need the knowledge of In order to ob
12、tain the following equation, we need8. 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 considerWithout loss of generality, we may assumeWithout loss of generality, we may prove the theorem in the caseIt suffices to prove the theore
13、m in the caseWe need only consider the caseFor simplicity, we may take9. 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
14、 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. 名师
15、资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 5 页，共 7 页 - - - - - - - - - 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) thenIf the assertion would not hold, thenThis is contrary toThis contradicts the fact t
16、hatThis 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 thatThus we conclude thatThus we are led to the
17、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名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 6 页，共 7 页 - - - - - - - - - We claim in factWe are now in a position toIf otherwiseProvided that 名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 7 页，共 7 页 - - - - - - - - -