<?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#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="frechet.html#V4">sequential</a>
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:39"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="frechet.html#V3">Frechet</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:39"/><i><font color="Green">E35</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="frechet.html#D5">FRECHET:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:39"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:39"/><i><font color="Green">E37</font></i>: 
for <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  <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> holds <br/> not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5:39"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E9:39"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E10:39"/>
for <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 <font color="Olive">b<sub>1</sub></font> is <a href="frechet.html#V2">convergent</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="frechet.html#K1">Lim</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="frechet.html#V2">convergent</a>
 <b>and </b><br/><a NAME="E2:39_1"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</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="E3:39_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E5:39_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E2:39_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="frechet2.html#E3:39_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="frechet2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:39_1"/>
<font color="Maroon">c<sub>5</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="txt" href="frechet2.html#E2:39"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:39"/><b>then </b>
 <a href="frechet2.html#K3" target="_self">Cl_Seq</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:39"><i><font color="Green">E33</font></i></a>, <a class="ref" href="frechet.html#D6">FRECHET:def 6</a></i>;<br/>
<a NAME="E12:39"/><b>then </b>
<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="txt" href="frechet2.html#E7:39"><i><font color="Green">E36</font></i></a>, <a class="txt" href="frechet2.html#E9:39"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pre_topc.html#T45">PRE_TOPC:45</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:39"/><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="frechet2.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b><a NAME="E15:39"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet2.html#E8:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="frechet2.html#E14:39"><i><font color="Green">E39</font></i></a></i>;<br/>


</div>
