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

<b>let </b><font color="Maroon">T</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:35"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">s1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> holds <br/>( <font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">s1</font></span>)</span> ) ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:35_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E2:35_1"/><i><font color="Green">E57</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"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> &amp; <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 ;<br/><a NAME="E3:35_1"/><b>then </b><i><font color="Green">E78</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/>;<br/><a NAME="E4:35_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E5:35_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#T39">NORMSP_1:39</a></i>;<br/><b>hence </b><a NAME="E6:35_1"/>
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="nfcont_1.html#K4" target="_self">*</a> <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> = 
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span>
<b>by </b><i>, <a class="ref" href="lopban_1.html#T47">LOPBAN_1:47</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="nfcont_1.html#K4" 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="E4:35"/>
<span class="p1"><a href="nfcont_1.html#K2" target="_self">||.</a><span class="default"><font color="Maroon">f</font></span><a href="nfcont_1.html#K2" target="_self">.||</a></span> <a href="nfcont_1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>

<a NAME="E5:35"/>
 <a href="vfunct_1.html#K6">-</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#T29">VFUNCT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/>
 <a href="vfunct_1.html#K6">-</a> <font color="Maroon">f</font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/>


</div>
