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数学专业英语-Phrases or clauses frequently used in mathematiscs

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发表于 2004-5-6 10:03:18 | 显示全部楼层 |阅读模式
< ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
< ><FONT size=3><FONT face="Times New Roman">I.</FONT>常见的涉及运算的短语或句子<FONT face="Times New Roman">(</FONT>句中<FONT face="Times New Roman">(A),(B)</FONT>表示某些表达式<FONT face="Times New Roman">,</FONT>如不等式<FONT face="Times New Roman">,</FONT>等式<FONT face="Times New Roman">,</FONT>方程等<FONT face="Times New Roman">)</FONT></FONT></P>
< ><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:path connecttype="rect" gradientshapeok="t"></v:path></v:shapetype><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1026"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">Differentiate</FONT></P>
<P ><FONT face="Times New Roman">Integrate</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3> 1.                    both sides of the equation and we get …   </FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1027"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">Differentiating </FONT></P>
<P ><FONT face="Times New Roman">Integrating </FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3> 2.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>                                          both sides of the equation, we get …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 3.Add (A) to (B) and we have …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 4.Substract (B) from (A) and we have …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 5.Multiplying each term of equation by …,we obtain …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 6.Dividing the equation through by …,we have…</FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1028"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">Give</FONT></P>
<P ><FONT face="Times New Roman">yield</FONT></P>
<P ><FONT face="Times New Roman">imply</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3> 7.(A) and (B) together                                  …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 8.Comparing (A) with (B) ,it is easy to see that …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 9.Substituting (A) into (B) ,we obtain …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 10.Eliminating (the parameter) from (A) and (B) ,we have …</FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1029"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">as follows </FONT></P>
<P ><FONT face="Times New Roman">in the following form</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3> 11.By introducing a new variable …,we can  then rewrite (A) </FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> 12.By a simple calculation,we obtain from (A)…</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">II.</FONT>定理证明过程中常见的短语和句子<FONT face="Times New Roman">.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>1.</FONT>       </FONT><FONT size=3>下面的句型可用来表达<FONT face="Times New Roman">”</FONT>根据什么即可得到什么<FONT face="Times New Roman">”</FONT>的意思<FONT face="Times New Roman">.</FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1030"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">definiton</FONT></P>
<P ><FONT face="Times New Roman">hypothesis</FONT></P>
<P ><FONT face="Times New Roman">assumptions</FONT></P>
<P ><FONT face="Times New Roman">theorem (N)</FONT></P>
<P ><FONT face="Times New Roman">lemma (A)</FONT></P>
<P ><FONT face="Times New Roman">corollary (B)</FONT></P>
<P ><FONT face="Times New Roman">the remark</FONT></P>
<P ><FONT face="Times New Roman">the fact that</FONT></P>
<P ><FONT face="Times New Roman">……………..</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman" size=3>According to                                     ,it follows …</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>Since …, it follows …</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>2.</FONT>       </FONT><FONT size=3>如果一个论断可以通过一些简单运算或简单推理而获得<FONT face="Times New Roman">,</FONT>由于这些运算或推理比较简单<FONT face="Times New Roman">,</FONT>读者可以自行推算<FONT face="Times New Roman">,</FONT>因而只需直接写出论断来<FONT face="Times New Roman">,</FONT>这时可用下面名句型<FONT face="Times New Roman">:</FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1031"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">see</FONT></P>
<P ><FONT face="Times New Roman">show </FONT></P>
<P ><FONT face="Times New Roman">prove </FONT></P>
<P ><FONT face="Times New Roman">verify </FONT></P>
<P ><FONT face="Times New Roman">check </FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman" size=3>It is easy to                                 that …</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1032"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">seen</FONT></P>
<P ><FONT face="Times New Roman">shown</FONT></P>
<P ><FONT face="Times New Roman">proved </FONT></P>
<P ><FONT face="Times New Roman">verified </FONT></P>
<P ><FONT face="Times New Roman">checked</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman" size=3>It can easily be                                       that …</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">3.</FONT>如果所要提及的结论比较显浅<FONT face="Times New Roman">,</FONT>或是众所周知<FONT face="Times New Roman">,</FONT>无需作过一步的证明<FONT face="Times New Roman">,</FONT>这时可用下面名句型<FONT face="Times New Roman">:</FONT></FONT></P>
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<P ><FONT face="Times New Roman">clear</FONT></P>
<P ><FONT face="Times New Roman">obvious</FONT></P>
<P ><FONT face="Times New Roman">evident</FONT></P>
<P ><FONT face="Times New Roman">well-known</FONT></P>
<P ><FONT face="Times New Roman"> <p></p></FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3> It is                                 that …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1034"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">Clerly</FONT></P>
<P ><FONT face="Times New Roman">Obviously</FONT></P>
<P ><FONT face="Times New Roman">Evidently</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3>                                         , …</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">4.</FONT>证明一个定理有时需要引进辅助函数<FONT face="Times New Roman">,</FONT>这时可用下面句型<FONT face="Times New Roman">:</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>Let us first define the fuction …</FONT></P>
<P ><FONT face="Times New Roman" size=3>Let us introduce a new function …</FONT></P>
<P ><FONT face="Times New Roman" size=3>Let us consider the function …</FONT></P>
<P ><FONT face="Times New Roman" size=3>Let us first investigate the function …</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> Or simply:</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>Let …</FONT></P>
<P ><FONT face="Times New Roman" size=3>Set …</FONT></P>
<P ><FONT face="Times New Roman" size=3>Define …</FONT></P>
<P ><FONT face="Times New Roman" size=3>Put …</FONT></P>
<P ><FONT face="Times New Roman" size=3>Consider …</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">5.</FONT>在一个定理中<FONT face="Times New Roman">,</FONT>有几个结论需要证明<FONT face="Times New Roman">,</FONT>其中有些结论比较明显<FONT face="Times New Roman">,</FONT>可不用证明<FONT face="Times New Roman">,</FONT>仅需证明余下结论便可<FONT face="Times New Roman">,</FONT>这时可用如下句型<FONT face="Times New Roman">:</FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1035"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">Obvious</FONT></P>
<P ><FONT face="Times New Roman">trivial</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3>  Since (A) and (B) are                                             ,we need only prove (C)</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>  Since (A) and (B) are trivial ,it suffices to prove (C)</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">6.</FONT>为了证明一个定理<FONT face="Times New Roman">,</FONT>有时我们并不是直接去证明<FONT face="Times New Roman">,</FONT>而是证明一个新的论断<FONT face="Times New Roman">,</FONT>一旦新的论断得到证明<FONT face="Times New Roman">,</FONT>已给定理不难由此而得证<FONT face="Times New Roman">,</FONT>这时可用下面句型<FONT face="Times New Roman">:</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1036"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape>
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<P ><FONT face="Times New Roman">Theorem</FONT></P>
<P ><FONT face="Times New Roman">result</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman" size=3>The                                 will be proved if we can show …</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>The theorem will be proved by showing that …</FONT></P>
<P ><FONT face="Times New Roman" size=3>If we can prove …then the theorem follows immediately </FONT></P>
<P ><FONT size=3>以下各句用于新的论断被证明之后<FONT face="Times New Roman">:</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>The theorem is now a direct consequence of what we have proved.</FONT></P>
<P ><FONT face="Times New Roman" size=3>The theorem follows immediately from what we have proved.</FONT></P>
<P ><FONT face="Times New Roman" size=3>The theorem is now evident from what we have proved.</FONT></P>
<P ><FONT face="Times New Roman" size=3>It is evident to see that the theorem holds .</FONT></P>
 楼主| 发表于 2004-5-6 10:03:52 | 显示全部楼层
< 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">7.</FONT>在证明过程中<FONT face="Times New Roman">,</FONT>有时要用到一些早已学过的知识或技巧<FONT face="Times New Roman">,</FONT>这里可用下面句子<FONT face="Times New Roman">,</FONT>以提醒读者<FONT face="Times New Roman">:</FONT></FONT></P>< 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> Recall that …</FONT></FONT></P>< 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> Notice that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> Note that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> Observe that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> In order to prove the theorem ,we need the knowledge of … (e.g ..the knowledge of Variations)</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> In order to obtain the following equation ,we need … (e.g .. Leibniz’s rule for differentiating under integralsign)</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">8.</FONT>如果需要证明的定理的假设条件是一般条件<FONT face="Times New Roman">,</FONT>但是<FONT face="Times New Roman">,</FONT>只要定理在特殊条件下成立<FONT face="Times New Roman">,</FONT>就不难推出定理在一般条件下也成立<FONT face="Times New Roman">,</FONT>这时仅需在特殊情况下去证明定理就够了<FONT face="Times New Roman">,</FONT>为此可用下面句型<FONT face="Times New Roman">:</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1037"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1037"><P 0cm 0cm 0pt"><FONT face="Times New Roman">Consider</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">assume</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3>  Without loss of generality ,we may                                 …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  Without loss generality ,we may prove the theorem in the case …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  It suffices to prove the theorem in the theorem in the case …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  We need only consider the case …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  For simplicity consider the case …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  (e.g .. we may take the domain to be the disk )</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt 258.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">9.</FONT>如果特征的论断可用以前用过的相似的方法或步骤进行证明<FONT face="Times New Roman">,</FONT>则可用下面句型<FONT face="Times New Roman">:</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 132.0pt 324.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1039"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1039"><P 0cm 0cm 0pt"><FONT face="Times New Roman">the same way</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">a similar way </FONT></P></DIV></TD></TR></TABLE><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1038"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1038"><P 0cm 0cm 0pt"><FONT face="Times New Roman">theorem</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">statement</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">lemma</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">…</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman"><FONT size=3> The                                    can be proved in                                        as shown before.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman" size=3> </FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> This theorem can be proved by the same method as employed in the last section.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> This theorem can be completed by the method analogous to that used above.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> Using the same argument as in the proof of theorem N, we can easily carry out the proof of this theorem.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> We now proceed as in the proof of theorem N.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> We shall adopt the same procedure as in the proof of theorem N.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman"><p></p></FONT></FONT> </P>
 楼主| 发表于 2004-5-6 10:04:04 | 显示全部楼层
< 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">10.</FONT>如果我们用的是反证法<FONT face="Times New Roman">,</FONT>则其开头及结尾可用下面句型<FONT face="Times New Roman">:</FONT></FONT></P>< 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1041"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1041">< 0cm 0cm 0pt"><FONT face="Times New Roman">false</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">not true</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">not right</FONT></P></DIV></TD></TR></TABLE><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1040"><FONT face="Times New Roman" size=3></FONT></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1040"><P 0cm 0cm 0pt"><FONT face="Times New Roman">statement</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">assertion</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">conclusion</FONT></P></DIV></TD></TR></TABLE><FONT face="Times New Roman" size=3>If the                                     were                                        then …</FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman" size=3>If the assertion would not hold ,then …</FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman" size=3>This is contrary to …</FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman" size=3>This contradicts the fact that ..</FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman" size=3>This leads to a contradiction.</FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">11.</FONT>表示定理已证毕或者把前面所证的总结为一结论<FONT face="Times New Roman">.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  We have thus proved the theorem .</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  This completes the proof .</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  The proof of  the theorem is now completed.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  It is now obvious that the theorem holds .</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  Thus we have derived that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  Consequently ,we infer that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  Thus we conclude that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  Thus we are led to the conclusion that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  Thus we arrive at the conclusion that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman"><FONT size=3>  Thus we can summarize what we have proved as the following theorem.</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">12.</FONT>其它</FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">  There exist (s) … such that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">  We claim … in fact …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">  We are now in a position to …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">  If otherwise …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT size=3><FONT face="Times New Roman">  Provided that …</FONT></FONT></P><P 0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><FONT face="Times New Roman" size=3>  </FONT></P><P 0cm 0cm 0pt"><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P><P 0cm 0cm 0pt"><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
发表于 2004-5-19 18:39:27 | 显示全部楼层
感谢分享![em07]
发表于 2004-12-19 06:28:50 | 显示全部楼层
<>很好!</P><>谢谢!</P>
发表于 2006-2-9 23:28:06 | 显示全部楼层
真好!谢谢!
发表于 2008-4-23 09:05:05 | 显示全部楼层
这个不错,学习了。
发表于 2009-12-16 15:48:39 | 显示全部楼层
不错,学习啊~~
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