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

<b>let </b><font color="Maroon">R</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be  <a href="wellfnd1.html#V2" target="_self">well_founded</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>, <font color="Maroon">Y</font> be  <a href="wellfnd1.html#V2" target="_self">well_founded</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>



<b>set </b><font color="Maroon">r</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>assume </b><a NAME="E1:20"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">X</font> is <a href="waybel_0.html#V12">lower</a>
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E28</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#R1">is_well_founded_in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20"/><i><font color="Green">E30</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#R1">is_well_founded_in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">XY</font> = <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E5:20"/>
<font color="Maroon">XY</font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a>,<a class="ref" href="wellfnd1.html#D3" target="_self">WELLFND1:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">XY</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>set </b><font color="Maroon">XZ</font> = <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font>;<br/>
<a NAME="E2:20_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1"/>
( <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1"/>
<font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:20_1_1_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2"/>
<font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_1_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/><b>take </b>
<font color="Maroon">a</font>
;<br/><b>thus </b><a NAME="E4:20_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E5:20_1_1_2"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:20_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:20_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:20_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E10:20_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:20_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E12:20_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E13:20_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:20"/>
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> is  <a href="wellfnd1.html#V2" target="_self">well_founded</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 ;<br/>


</div>
