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

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Vt</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vt</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">V</font> is <a href="nfcont_1.html#V2">closed</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">St</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">St</font> is <a href="frechet.html#V2">convergent</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">St</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Vt</font> )
 ;<br/><b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">St</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:23_1_1"/>
<font color="Maroon">S</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_1_1"/><b>then </b>
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D5">NFCONT_1:def 5</a>, , , </i>;<br/><a NAME="E5:23_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">S</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:23_1_1"/>
 <a href="frechet.html#K1">Lim</a> <font color="Maroon">St</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Vt</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1"/>
<font color="Maroon">Vt</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="frechet.html#D6">FRECHET:def 6</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:23"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Vt</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:23_2"/><i><font color="Green">E44</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">S</font> is <a href="normsp_1.html#V4">convergent</a> )
 ;<br/><b>reconsider </b><font color="Maroon">St</font> = <font color="Maroon">S</font> as   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> ;<br/><a NAME="E3:23_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">St</font> is <a href="frechet.html#V2">convergent</a>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_2"/><b>then </b>
 <a href="frechet.html#K1">Lim</a> <font color="Maroon">St</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Vt</font>
 <b>by </b><i>, , <a class="ref" href="frechet.html#D6">FRECHET:def 6</a>, </i>;<br/><a NAME="E5:23_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">S</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:23_2"/>
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:23"/>
<font color="Maroon">V</font> is <a href="nfcont_1.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D5">NFCONT_1:def 5</a></i>;<br/>


</div>
