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<b>let </b><font color="Maroon">c<sub>1</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="E1:76"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cfcont_1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cfcont_1.html#R2" target="_self">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:76_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:76_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E3:76_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:76_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E4:76_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:76_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E4:76_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="cfcont_1.html#S1" target="_self">CFCONT_1:sch 1</a>(<a class="txt" href="cfcont_1.html#E2:76_1"><i><font color="Green">E52</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:76_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:76_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E3:76_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:76_1"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:76_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:76_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:76_1"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E2:76_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><a NAME="E8:76_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E10:76_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is    <a href="comseq_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>3</sub></font> &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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:76"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cfcont_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E11:76_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:76"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cfcont_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E12:76_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E13:76_1"/>
 <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#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cfcont_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E14:76_1"/><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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:76_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <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; <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></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">c<sub>1</sub></font> <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#E10:76_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cfcont_1.html#E12:76_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E16:76_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <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; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <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#E10:76_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E17:76_1"/>
( <font color="Maroon">c<sub>5</sub></font> is    <a href="comseq_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</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>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cfcont_1.html#E8:76_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cfcont_1.html#E16:76_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cfcont_1.html#T26" target="_self">Th26</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:76"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cfcont_1.html#V1" target="_self">compact</a>
 <b>by </b><i><a class="ref" href="cfcont_1.html#D6" target="_self">Def6</a></i>;<br/>


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