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<b>let </b><font color="Maroon">seq'</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq'</font> is <a href="seq_2.html#V4" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E39</font></i>: 
 <a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 ;<br/>

<a NAME="E5:31"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4" target="_self">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T3" target="_self">Th3</a>, <a class="txt" href="seq_2.html#E4"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="seq_2.html#E5"><i><font color="Green">Lemma34</font></i></a>, </i>;<br/>
<b>then </b> <a href="seq_2.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq'</font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq'</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq'</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="seq_2.html#T3" target="_self">Th3</a>, <a class="txt" href="seq_2.html#E4"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="seq_2.html#E5"><i><font color="Green">Lemma34</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq'</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E7:31"/>
 <a href="seq_2.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq'</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq'</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>;<br/>


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