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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st 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" title="b1">x</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> holds <br/><font color="Olive" title="b1">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></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 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="c2">x</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> implies <font color="Maroon" title="c2">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> )</span><br/>



<b>set </b><font color="Maroon" title="c3">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="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/>
<b>set </b><font color="Maroon" title="c5">xwfp</font> = <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E1:21"/><i><font color="Green" title="E15">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/>
<a NAME="E2:21"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:21"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">xwfp</font> = <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>assume </b><a NAME="E5:21"/>
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="c2">x</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">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></span><br/>

<a NAME="E6:21"/><b>then </b><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c6">xwfp</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T6" target="_self" title="WELLFND1:th.6">Th6</a></i>;<br/>
<a NAME="E7:21"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T7" target="_self" title="WELLFND1:th.7">Th7</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
<font color="Maroon" title="c6">xwfp</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T8" target="_self" title="WELLFND1:th.8">Th8</a></i>;<br/>
<a NAME="E9:21"/><b>then </b>
<font color="Maroon" title="c6">xwfp</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#E6:21"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:21"/>
<font color="Maroon" title="c2">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>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="wellfnd1.html#E3:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
