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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel18.html#V1">TopSpace-yielding</a>  <a href="pboole.html#M1">ManySortedSet</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K2">product_prebasis</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:23"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> holds not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>



<b>assume </b><a NAME="E2:23"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel18.html#K2">product_prebasis</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:23"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E1:23"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow17.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E7:23"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="yellow17.html#E5:23"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:23_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:23_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow17.html#E1:23"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:23"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:23"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E11:23"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E12:23"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E4:23"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow17.html#E6:23"><i><font color="Green">E26</font></i></a>, <a class="ref" href="yellow17.html#T18" target="_self">Th18</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E13:23"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow17.html#E9:23"><i><font color="Green">E27</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:23"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E10:23"><i><font color="Green">E28</font></i></a></i>;<br/>

<b>thus </b><a NAME="E15:23"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E5:23"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow17.html#E12:23"><i><font color="Green">E30</font></i></a></i>;<br/>

<b>thus </b><a NAME="E16:23"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow17.html#E11:23"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
