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

<b>set </b><font color="Maroon" title="c2">wfp</font> =  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c3">r</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">IT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span> ;<br/>
<a NAME="E1:17_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
 <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D4" target="_self" title="WELLFND1:def.4">Def4</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D1" target="_self" title="WELLFND1:def.1">WELLFND1:def 1</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:17_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:17_1_1_1"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c8">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Maroon" title="c8">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )
 <b>by </b><i><a class="txt" href="wellfnd1.html#E3:17_1_1_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/><a NAME="E6:17_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:17_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="wellfnd1.html#E3:17_1_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="wellfnd1.html#E5:17_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="wellfnd1.html#D1" target="_self" title="WELLFND1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E7:17_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:17_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="wellfnd1.html#E3:17_1_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="wellfnd1.html#E5:17_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a>,<a class="ref" href="wellfnd1.html#D3" target="_self" title="WELLFND1:def.3">WELLFND1:def 3</a><br/>

<b>assume </b><a NAME="E3:17_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:17_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:17_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">YY</font> &amp; <font color="Maroon" title="c7">YY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:17_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="wellfnd1.html#E3:17_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="wellfnd1.html#E5:17_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E9:17_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
( <font color="Maroon" title="c7">YY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c8">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Maroon" title="c8">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )
 <b>by </b><i><a class="txt" href="wellfnd1.html#E7:17_1_1"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/>
<a NAME="E10:17_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> <font color="Maroon" title="c8">S</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E9:17_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="wellfnd1.html#D3" target="_self" title="WELLFND1:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">YS</font> = <font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font>;<br/>
<a NAME="E11:17_1_1"/>
( <font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E5:17_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="wellfnd1.html#E7:17_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="wellfnd1.html#E9:17_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:17_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font> )
 <b>by </b><i><a class="txt" href="wellfnd1.html#E10:17_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E14:17_1_1"/><i><font color="Green" title="E18">A11</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E13:17_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E19">A12</font></i>: <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c5">Y</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E9:17_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="wellfnd1.html#T5" target="_self" title="WELLFND1:th.5">Th5</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
;<br/>
<a NAME="E16:17_1_1"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E13:17_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E17:17_1_1"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E15:17_1_1"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E18:17_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="wellfnd1.html#E14:17_1_1"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/>


</div>
