<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</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>thus </b><a NAME="E1:62"/>
( <font color="Maroon">f</font> <a href="cfcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">X</font> implies ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</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> <font color="Maroon">X</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#R2">in</a> <font color="Maroon">X</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> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Olive">s1</font></span>)</span> <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> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="cfcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:62_1"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Olive">x1</font> ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:62_1_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> <font color="Maroon">X</font> &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#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><i><font color="Green">E69</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> = 
<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#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E3:62_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">s1</font>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:62_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/><a NAME="E2:62_1_1_2"/><i><font color="Green">E72</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>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<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#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E5:62_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
 <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/><b>hence </b><a NAME="E7:62_1_1"/>
( <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> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62_1"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</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> <font color="Maroon">X</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#R2">in</a> <font color="Maroon">X</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> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Olive">s1</font></span>)</span> <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>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:62"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</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> <font color="Maroon">X</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#R2">in</a> <font color="Maroon">X</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> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Olive">s1</font></span>)</span> <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> ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><i><font color="Green">E88</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> = 
<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#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:62_2_2"/><i><font color="Green">E89</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="relset_1.html#K8">|</a> <font color="Maroon">X</font></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">x1</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:62_2_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/><a NAME="E2:62_2_2_1"/><i><font color="Green">E91</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>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<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#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E3:62_2_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="partfun2.html#T32">PARTFUN2:32</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="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, , , , </i>
;<br/><b>hence </b><a NAME="E5:62_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></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> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <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="relset_1.html#K8">|</a> <font color="Maroon">X</font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:62_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62"/>
<font color="Maroon">f</font> <a href="cfcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div>
