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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">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><a NAME="E1:65"/><i><font color="Green" title="E44">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#R2" title="CFCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E2:65"/><b>then </b>
<font color="Maroon" title="c1">X</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="c3">f</font>
 
<b>by </b><i><a class="ref" href="cfcont_1.html#D5" target="_self" title="CFCONT_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:65"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c2">X1</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="c3">f</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:65"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cfcont_1.html#D5" target="_self" title="CFCONT_1:def.5">CFCONT_1:def 5</a><br/>

<b>let </b><font color="Maroon" title="c4">g</font> be    <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> ;<br/>

<b>assume </b><a NAME="E4:65"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font>
 ;<br/>

<a NAME="E5:65"/><b>then </b><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:65"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="cfcont_1.html#D5" target="_self" title="CFCONT_1:def.5">Def5</a></i>;<br/>
<b>thus </b><a NAME="E6:65"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<a NAME="E1:65_1"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">X1</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:65"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="cfcont_1.html#E4:65"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:65_1"/><i><font color="Green" title="E48">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cfcont_1.html#D2" target="_self" title="CFCONT_1:def.2">CFCONT_1:def 2</a><br/>

<b>let </b><font color="Maroon" title="c5">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E3:65_1"/><i><font color="Green" title="E49">A6</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">s1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> &amp; <font color="Maroon" title="c5">s1</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">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> )
 ;<br/>

<a NAME="E4:65_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:65"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E5:65_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:65_1"/><b>then </b><i><font color="Green" title="E50">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:65_1"/><b>then </b><i><font color="Green" title="E51">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">s1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:65_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:65_1"/><b>then </b><i><font color="Green" title="E52">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c5">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">g</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E5:65"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cfcont_1.html#E3:65_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self" title="CFCONT_1:def.2">Def2</a></i>;<br/>
<i><font color="Green" title="E53">A10</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">g</font> = 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">g</font>
<b>by </b><i><a class="txt" href="cfcont_1.html#E2:65_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="cfcont_1.html#E6:65_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">g</font>
<b>by </b><i><a class="txt" href="cfcont_1.html#E2:65_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="E1:65_1_2"/><i><font color="Green" title="E54">A11</font></i>: 
<font color="Maroon" title="c5">s1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">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="c5">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>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E7:65_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self" title="CFCONT_1:th.16">Th16</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E7:65_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="cfcont_1.html#E1:65_1_2"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:65_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="cfcont_1.html#E1:65_1_2"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="txt" href="cfcont_1.html#E3:65_1"><i><font color="Green" title="E49">A6</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>
<b>hence </b><a NAME="E11:65_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c5">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">g</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X1</font></span>)</span> <a href="cfcont_1.html#K1" title="CFCONT_1:func.1">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E8:65_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="cfcont_1.html#E9:65_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="comseq_1.html#T6" title="COMSEQ_1:th.6">COMSEQ_1:6</a></i>;<br/>


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


</div>
