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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">T</font> is <a href="frechet.html#V3" target="_self">Frechet</a>
 ;<br/>

<a NAME="E2:36"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st ( for <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font>  st <font color="Olive">S</font> is <a href="frechet.html#V2" target="_self">convergent</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> holds <br/> <a href="frechet.html#K1" target="_self">Lim</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> ) holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font>  st <font color="Olive">S</font> is <a href="frechet.html#V2" target="_self">convergent</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> holds <br/> <a href="frechet.html#K1" target="_self">Lim</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:36_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E3:36_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:36_1_1"/><i><font color="Green">E40</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">A</font>
 <b>and </b><br/><a NAME="E5:36_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1" target="_self">Lim</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="frechet.html#E6"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="frechet.html#T5" target="_self">Th5</a>, <a class="ref" href="frechet.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:36_1_1"/>
<font color="Maroon">S</font> <a href="frechet.html#R1" target="_self">is_convergent_to</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E2:36_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E7:36_1_1"/><b>then </b>
<font color="Maroon">S</font> is <a href="frechet.html#V2" target="_self">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:36_1_1"/><b>then </b>
 <a href="frechet.html#K1" target="_self">Lim</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:36_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="frechet.html#E2:36_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:36_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="frechet.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
<font color="Maroon">T</font> is <a href="frechet.html#V4" target="_self">sequential</a>
 <b>by </b><i/>;<br/>


</div>
