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<b>let </b><font color="Maroon">x0</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">f</font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">s1</font> being  <a href="comseq_1.html#NM1">Complex_Sequence</a>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">s1</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> holds <br/>( <font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Olive">s1</font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Olive">s1</font></span>)</span> ) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:59"/>
not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:59"/><b>then </b>
not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="partfun2.html#T44">PARTFUN2:44</a></i>;<br/>
<a NAME="E5:59"/><b>then </b>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:59"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2">^</a> </span>)</span>
 <b>by </b><i><a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cfcont_1.html#D2" target="_self">CFCONT_1:def 2</a><br/>

<a NAME="E7:59"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>let </b><font color="Maroon">s1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E8:59"/><i><font color="Green">E70</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2">^</a> </span>)</span> &amp; <font color="Maroon">s1</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 ;<br/>

<a NAME="E9:59"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>;<br/>
<a NAME="E10:59"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:59"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:59"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font> is <a href="comseq_1.html#V1">non-zero</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:59"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="comseq_1.html#K7">"</a>  is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i>, , <a class="ref" href="comseq_2.html#T34">COMSEQ_2:34</a></i>;<br/>
<b>hence </b><a NAME="E14:59"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font> is <a href="comseq_2.html#V2">convergent</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i>, <a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>
<br/>.= 
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="comseq_1.html#K7">"</a> </span>)</span>
<b>by </b><i>, , , <a class="ref" href="comseq_2.html#T35">COMSEQ_2:35</a></i>
<br/>.= 
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


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