<?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#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">S1</font> be    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="frechet2.html#R1" target="_self">is_a_cluster_point_of</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">n0</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n0</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">n0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:61"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font></span>)</span>
 <b>by </b><i><a class="ref" href="frechet2.html#T3" target="_self">Th3</a></i>;<br/>
<b>let </b><font color="Maroon">O</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet2.html#D4" target="_self">FRECHET2:def 4</a><br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E5:61"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">O</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:61"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/>

<b>consider </b><font color="Maroon">m0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:61"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m0</font>
 <b>and </b><br/><a NAME="E9:61"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E11"><i><font color="Green">Lemma48</font></i></a>, , , <a class="txt" href="frechet2.html#E5"><i><font color="Green">Lemma37</font></i></a></i>;<br/>
<a NAME="E10:61"/>
<font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E11:61"/><b>then </b>
<font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m0</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E5:61"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E13:61"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5:61"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>

<font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n0</font></span>)</span>
<b>by </b><i><a class="ref" href="frechet2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m0</font>

;<br/>
<b>hence </b><a NAME="E15:61"/>
<font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E14"><i><font color="Green">Lemma54</font></i></a></i>;<br/>


</div>
