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

<a NAME="E1:41_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel30.html#K1" target="_self">^0</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel30.html#K1" target="_self">^0</a>  <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1"><b>proof </b></a><div class="add">

<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><b>that </b><br/><a NAME="E1:41_1_1_1_1"/><i><font color="Green">E21</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> <a href="waybel30.html#K1" target="_self">^0</a> 
 <b>and </b><br/><a NAME="E2:41_1_1_1_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:41_1_1_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel30.html#E1:41_1_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <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> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_0.html#T5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E6:41_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<a NAME="E7:41_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/>
<b>hence </b><a NAME="E8:41_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel30.html#E4:41_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel30.html#E6:41_1_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:41_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel30.html#K1" target="_self">^0</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel30.html#K1" target="_self">^0</a>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div>
