<?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>set </b><font color="Maroon">r</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">c</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">wfp</font> =  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font>;<br/>
<a NAME="E1:21"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">cs</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<b>set </b><font color="Maroon">IT</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : ( <font color="Olive">S</font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V12">lower</a> ) </span>}</span> ;<br/>
<a NAME="E3:21"/><i><font color="Green">E26</font></i>: 
 <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : ( <font color="Olive">S</font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V12">lower</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">R</font> is <a href="wellfnd1.html#V1" target="_self">well_founded</a>
 ;<br/><a NAME="E2:21_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">cs</font> is <a href="waybel_0.html#V12">lower</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D1" target="_self">WELLFND1:def 1</a><br/>

<b>thus </b><a NAME="E1:21_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cs</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</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> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cs</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:21_1"/>
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">cs</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:21_1"/><b>then </b>
<font color="Maroon">cs</font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a>
 <b>by </b><i/>;<br/><a NAME="E5:21_1"/><b>then </b>
<font color="Maroon">cs</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : ( <font color="Olive">S</font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V12">lower</a> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:21_1"/><b>then </b>
<font color="Maroon">cs</font> <a href="tarski.html#R1">c=</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E7:21_1"/>
 <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:21"/><i><font color="Green">E33</font></i>: 
 <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>let </b><font color="Maroon">Y</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#D2" target="_self">WELLFND1:def 2</a><br/>

<b>assume </b><a NAME="E6:21"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:21"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">YY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:21"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font> &amp; <font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : ( <font color="Olive">S</font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V12">lower</a> ) </span>}</span>  )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E12:21"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">YY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S</font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V12">lower</a> )
 <b>by </b><i/>;<br/>
<a NAME="E13:21"/><i><font color="Green">E68</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">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">YS</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>;<br/>
<a NAME="E14:21"/>
( <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:21"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</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">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E17:21"/><i><font color="Green">E72</font></i>: 
<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E18:21"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:21"/><b>then </b>
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="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><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">Y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<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="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<a NAME="E21:21"/><b>then </b>
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">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E22:21"/>
 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">Y</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">Y</font> )
 <b>by </b><i/>;<br/>


</div>
