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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> , <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:76"/><i><font color="Green" title="E49">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> is <a href="cfcont_1.html#V1" title="CFCONT_1:attr.1">compact</a>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</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" title="c2">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E51">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/><a NAME="E2:76_1"/><i><font color="Green" title="E52">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">g</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" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:76_1_1"/>
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s1</font>
 
<b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self" title="CFCONT_1:th.5">Th5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E3:76_1_1"/><i><font color="Green" title="E53">A5</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">g</font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:76_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">g</font>
;<br/>

<b>thus </b><a NAME="E4:76_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">g</font>]
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:76_1_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c3">q1</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><b> such that </b><br/><a NAME="E4:76_1"/><i><font color="Green" title="E53">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">q1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="cfcont_1.html#S1" target="_self" title="CFCONT_1:sch.1">CFCONT_1:sch 1</a>(<a class="txt" href="cfcont_1.html#E2:76_1"><i><font color="Green" title="E52">A4</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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:76_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">q1</font>
 ;<br/><a NAME="E2:76_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self" title="CFCONT_1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E3:76_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:76_1"><i><font color="Green" title="E53">A6</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:76_1"/><b>then </b><i><font color="Green" title="E54">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:76_1_3"/>
( <font color="Maroon" title="c3">q1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:76_1"><i><font color="Green" title="E53">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E2:76_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c3">q1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self" title="CFCONT_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div><a NAME="E8:76_1"/><b>then </b><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c3">q1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">s1</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6" title="COMSEQ_1:th.6">COMSEQ_1:6</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">s2</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><b> such that </b><br/><a NAME="E10:76_1"/><i><font color="Green" title="E56">A9</font></i>: 
( <font color="Maroon" title="c4">s2</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">q1</font> &amp; <font color="Maroon" title="c4">s2</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:76"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="cfcont_1.html#D6" target="_self" title="CFCONT_1:def.6">Def6</a></i>;<br/><a NAME="E11:76_1"/>
<font color="Maroon" title="c1">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:76"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="cfcont_1.html#D5" target="_self" title="CFCONT_1:def.5">Def5</a></i>;<br/><a NAME="E12:76_1"/><b>then </b><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/><a NAME="E13:76_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">q1</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="cfcont_1.html#T7" target="_self" title="CFCONT_1:th.7">Th7</a></i>;<br/><a NAME="E14:76_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:76_1"/><b>then </b>
( <font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s2</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="cfcont_1.html#E12:76_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self" title="CFCONT_1:def.2">Def2</a></i>;<br/><a NAME="E16:76_1"/><b>then </b><i><font color="Green" title="E58">A11</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s2</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>take </b><font color="Maroon" title="c5">q2</font> = <font color="Maroon" title="c1">f</font> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c4">s2</font>;<br/><b>thus </b><a NAME="E17:76_1"/>
( <font color="Maroon" title="c5">q2</font> is    <a href="comseq_3.html#M1" title="COMSEQ_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">s1</font> &amp; <font color="Maroon" title="c5">q2</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:76_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="cfcont_1.html#E8:76_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="cfcont_1.html#E10:76_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="cfcont_1.html#E16:76_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="cfcont_1.html#T26" target="_self" title="CFCONT_1:th.26">Th26</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font> is <a href="cfcont_1.html#V1" title="CFCONT_1:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="cfcont_1.html#D6" target="_self" title="CFCONT_1:def.6">Def6</a></i>;<br/>


</div>
