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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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><b>that </b><br/><a NAME="E1:71"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#R2" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<i><font color="Green">E49</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E2:71"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>

;<br/>
<b>hence </b><a NAME="E4:71"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:71"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cfcont_1.html#D5" target="_self">Def5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cfcont_1.html#D5" target="_self">CFCONT_1:def 5</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E5:71"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E6:71"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:71"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cfcont_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E7:71"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E4:71"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cfcont_1.html#E5:71"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:71"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:71_2"/><i><font color="Green">E54</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:71_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E3:71_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:71"><i><font color="Green">E49</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E4:71_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:71"><i><font color="Green">E52</font></i></a>, <a class="txt" href="cfcont_1.html#E1:71_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:71_2"/>
<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:71"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cfcont_1.html#E4:71"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cfcont_1.html#E5:71"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:71_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E7:71_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>;<br/><div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:71_2_1"/><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:71"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cfcont_1.html#E1:71_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E2:71_2_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E3:71_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:71_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:71_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self">Th5</a></i>;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><a NAME="E2:71_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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/><b>hence </b><a NAME="E3:71_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:71"><i><font color="Green">E48</font></i></a>, <a class="txt" href="cfcont_1.html#E3:71_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="cfcont_1.html#E4:71_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="partfun2.html#T44">PARTFUN2:44</a></i>;<br/></div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:71_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:71_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="cfcont_1.html#E4:71_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
;<br/><b>hence </b><a NAME="E7:71_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="txt" href="cfcont_1.html#E5:71_2_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><a NAME="E2:71_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></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/><b>hence </b><a NAME="E3:71_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:71"><i><font color="Green">E48</font></i></a>, <a class="txt" href="cfcont_1.html#E3:71"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cfcont_1.html#E4:71"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cfcont_1.html#E5:71"><i><font color="Green">E51</font></i></a>, <a class="ref" href="partfun2.html#T44">PARTFUN2:44</a></i>;<br/></div><b>end;</b></div><a NAME="E10:71_2"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_1.html#V1">non-zero</a> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:71"><i><font color="Green">E52</font></i></a>, <a class="txt" href="cfcont_1.html#E1:71_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cfcont_1.html#E3:71_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="cfcont_1.html#E7:71_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="cfcont_1.html#E8:71_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self">Def2</a>, <a class="ref" href="comseq_1.html#T4">COMSEQ_1:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:71_2_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E2:71_2_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:71_2"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E3:71_2_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E4:71_2_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:71_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:71_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cfcont_1.html#E1:71_2_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E4:71_2_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:71_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="cfcont_1.html#E1:71_2_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:71_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K7">"</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="comseq_1.html#D11">COMSEQ_1:def 11</a></i>
;<br/></div><b>end;</b></div><a NAME="E12:71_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K7">"</a> 
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/><b>hence </b><a NAME="E13:71_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:71_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cfcont_1.html#E10:71_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="comseq_2.html#T34">COMSEQ_2:34</a></i>;<br/><b>thus </b> <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E4:71_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cfcont_1.html#E10:71_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="cfcont_1.html#E12:71_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="comseq_2.html#T35">COMSEQ_2:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K12">"</a> 
<b>by </b><i><a class="txt" href="cfcont_1.html#E6:71_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="cfcont_1.html#E4:71"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cfcont_1.html#E5:71"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cfunct_1.html#D2">CFUNCT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="cfcont_1.html#E8:71"><i><font color="Green">E53</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:71"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfunct_1.html#K2">^</a> </span>)</span> <a href="relset_1.html#K8">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E8:71"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self">Def2</a></i>;<br/>


</div>
