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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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="E1:35"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:35"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>take </b>
<a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E1:35_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></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="E1:35_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:35_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>

<a NAME="E3:35_1"/>
<a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="frechet.html#T23">FRECHET:23</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="frechet2.html#D2" target="_self">Def2</a></i>;<br/>


</div>
