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<b>let </b><font color="Maroon">CNS1</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">CNS2</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">CNS1</font>,<font color="Maroon">CNS2</font>;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>;<br/>







<b>assume </b><a NAME="E1:79"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:79"/><b>then </b><i><font color="Green">E35</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="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font>  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="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">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="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">s1</font> is <a href="clvect_1.html#V6">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="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Olive">s1</font></span>)</span> ) ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:79_1"/><i><font color="Green">E54</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"><a href="ncfcont1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K2" target="_self">.||</a></span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D20" target="_self">Def20</a>, </i>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font>;<br/><b>assume </b><a NAME="E2:79_1"/><i><font color="Green">E56</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"><a href="ncfcont1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K2" target="_self">.||</a></span> &amp; <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 ;<br/><a NAME="E3:79_1"/><b>then </b><i><font color="Green">E109</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> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E4:79_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">f</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">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="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#D19" target="_self">Def19</a>, , </i>;<br/><a NAME="E5:79_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="clvect_1.html#T119">CLVECT_1:119</a></i>;<br/><b>hence </b><a NAME="E6:79_1"/>
<span class="p1"><a href="ncfcont1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K2" target="_self">.||</a></span> <a href="ncfcont1.html#K10" target="_self">*</a> <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T6" target="_self">Th6</a>, <a class="ref" href="ncfcont1.html#D13" target="_self">Def13</a></i>;<br/><b>thus </b><span class="p1"><a href="ncfcont1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> = 
<span class="p1"><a href="ncfcont1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="ncfcont1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K5" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T7" target="_self">Th7</a>, <a class="ref" href="clopban1.html#T45">CLOPBAN1:45</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K2" target="_self">.||</a></span> <a href="ncfcont1.html#K10" target="_self">*</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T6" target="_self">Th6</a>, <a class="ref" href="ncfcont1.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:79"/>
<span class="p1"><a href="ncfcont1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="ncfcont1.html#K2" target="_self">.||</a></span> <a href="ncfcont1.html#R5" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>

<a NAME="E5:79"/>
 <a href="vfunct_2.html#K6">-</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="vfunct_2.html#K4">(#)</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="vfunct_2.html#T23">VFUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E6:79"/>
 <a href="vfunct_2.html#K6">-</a> <font color="Maroon">f</font> <a href="ncfcont1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D19" target="_self">Def19</a>, <a class="ref" href="ncfcont1.html#D16" target="_self">Def16</a></i>;<br/>


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