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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">C</font> is    <a href="taxonom1.html#M2" title="TAXONOM1:mode.2">Strong_Classification</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">B</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font>;<br/>
<a NAME="E2:9"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="E3:9"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="taxonom1.html#D2" title="TAXONOM1:def.2">TAXONOM1:def 2</a></i>;<br/>
<a NAME="E4:9"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</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="taxonom2.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">R'</font> =  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">R</font> =  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font> ;<br/>
<a NAME="E6:9"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1" title="YELLOW_1:def.1">YELLOW_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">D</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #);<br/>
<a NAME="E7:9"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">C</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="taxonom1.html#D2" title="TAXONOM1:def.2">TAXONOM1:def 2</a></i>;<br/>
<a NAME="E8:9"/><i><font color="Green" title="E11">A6</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="taxonom1.html#D2" title="TAXONOM1:def.2">TAXONOM1:def 2</a></i>;<br/>
<a NAME="E9:9"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">B'</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="taxonom2.html#E8:9"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:9"/><i><font color="Green" title="E13">A8</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #) is <a href="taxonom2.html#V1" title="TAXONOM2:attr.1">with_superior</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c1">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #) <b>by </b><i><a class="txt" href="taxonom2.html#E8:9"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="taxonom2.html#E9:9"><i><font color="Green" title="E12">A7</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="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">SmallestPartition</a> <font color="Maroon" title="c1">X</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>reconsider </b><font color="Maroon" title="c10">C'</font> = <font color="Maroon" title="c2">C</font> as    <a href="taxonom1.html#M2" title="TAXONOM1:mode.2">Strong_Classification</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<a NAME="E5:9_1"/>
 <a href="eqrel_1.html#K11" title="EQREL_1:func.11">SmallestPartition</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C'</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#D2" title="TAXONOM1:def.2">TAXONOM1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #) <b>by </b><i><a class="txt" href="taxonom2.html#E3:9_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">s</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom2.html#D1" target="_self" title="TAXONOM2:def.1">TAXONOM2:def 1</a><br/>

<a NAME="E7:9_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c8">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E4:9"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="taxonom2.html#E3:9_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E8:9_1"/><b>then </b><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">R</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green" title="E16">A11</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">R</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">s</font> )
 ;<br/><a NAME="E2:9_1_1"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9_1_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="relat_1.html#D6" title="RELAT_1:def.6">RELAT_1:def 6</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_1"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">R</font> or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">R</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E2:9_1_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">R</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> ;<br/><a NAME="E3:9_1_1_1_1"/>
<font color="Maroon" title="c13">y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:9"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="taxonom2.html#T4" target="_self" title="TAXONOM2:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c8">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">R</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> ;<br/><a NAME="E3:9_1_1_1_2"/>
<font color="Maroon" title="c13">y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:9"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="taxonom2.html#T4" target="_self" title="TAXONOM2:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c8">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:9_1"/>
<font color="Maroon" title="c8">s</font> <a href="orders_1.html#R8" title="ORDERS_1:pred.8">is_superior_of</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #)
 <b>by </b><i><a class="txt" href="taxonom2.html#E8:9_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="orders_1.html#D13" title="ORDERS_1:def.13">ORDERS_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #);<br/><b>given </b><font color="Maroon" title="c10">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c5">R</font> #)<b> such that </b><a NAME="E1:9_2"/><i><font color="Green" title="E14">A13</font></i>: 
( <font color="Maroon" title="c10">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c10">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">z'</font> = <font color="Maroon" title="c10">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> ;<br/><b>consider </b><font color="Maroon" title="c12">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:9_2"/><i><font color="Green" title="E15">A14</font></i>: 
( <font color="Maroon" title="c11">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Z</font> &amp; <font color="Maroon" title="c12">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">z''</font> = <font color="Maroon" title="c11">z'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom2.html#E7:9"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="taxonom2.html#T4" target="_self" title="TAXONOM2:th.4">Th4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">Z'</font> = <font color="Maroon" title="c12">Z</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="taxonom2.html#E4:9_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><a NAME="E7:9_2"/>
<font color="Maroon" title="c13">z''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Z'</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:9_2"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/><a NAME="E8:9_2"/><b>then </b>
<font color="Maroon" title="c13">z''</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="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:9_2"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">z</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/><a NAME="E11:9_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E12:9_2"/><b>then </b><i><font color="Green" title="E17">A16</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:9"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/><a NAME="E13:9_2"/><b>then </b><i><font color="Green" title="E18">A17</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green" title="E16">A15</font></i></a></i>;<br/><a NAME="E14:9_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E15:9_2"/><b>then </b><i><font color="Green" title="E19">A18</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:9"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/><a NAME="E16:9_2"/><b>then </b><i><font color="Green" title="E20">A19</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green" title="E16">A15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c17">y'</font> = <font color="Maroon" title="c9">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">B'</font> ;<br/><b>consider </b><font color="Maroon" title="c18">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:9_2"/><i><font color="Green" title="E21">A20</font></i>: 
( <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">S</font> &amp; <font color="Maroon" title="c18">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">S'</font> = <font color="Maroon" title="c18">S</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">T</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E22:9_2"/><i><font color="Green" title="E22">A21</font></i>: 
( <font color="Maroon" title="c17">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">T</font> &amp; <font color="Maroon" title="c20">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">T'</font> = <font color="Maroon" title="c20">T</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/><a NAME="E24:9_2"/><i><font color="Green" title="E23">A22</font></i>: 
<font color="Maroon" title="c2">C</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="taxonom1.html#D2" title="TAXONOM1:def.2">TAXONOM1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_1_1"/>
( <font color="Maroon" title="c19">S'</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">T'</font> or <font color="Maroon" title="c21">T'</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">S'</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="taxonom2.html#E24:9_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_1_1"/>
<font color="Maroon" title="c19">S'</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">T'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c22">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_2_1_1_1"/><i><font color="Green" title="E24">A23</font></i>: 
( <font color="Maroon" title="c22">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">T'</font> &amp; <font color="Maroon" title="c16">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">Y</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/><b>thus </b><a NAME="E4:9_2_1_1_1"/>
( <font color="Maroon" title="c16">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">y'</font> or <font color="Maroon" title="c17">y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">x'</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="taxonom2.html#E13:9_2"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="taxonom2.html#E15:9_2"><i><font color="Green" title="E19">A18</font></i></a>, <a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="taxonom2.html#E3:9_2_1_1_1"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="taxonom2.html#T3" target="_self" title="TAXONOM2:th.3">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_1_2"/>
<font color="Maroon" title="c21">T'</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">S'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c22">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_2_1_1_2"/><i><font color="Green" title="E24">A24</font></i>: 
( <font color="Maroon" title="c22">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">S'</font> &amp; <font color="Maroon" title="c17">y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">Y</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/><b>thus </b><a NAME="E4:9_2_1_1_2"/>
( <font color="Maroon" title="c16">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">y'</font> or <font color="Maroon" title="c17">y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">x'</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="taxonom2.html#E12:9_2"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="taxonom2.html#E16:9_2"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="taxonom2.html#E3:9_2_1_1_2"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="taxonom2.html#T3" target="_self" title="TAXONOM2:th.3">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E26:9_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">x'</font>,<font color="Maroon" title="c17">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">y'</font>,<font color="Maroon" title="c16">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> <font color="Maroon" title="c5">R</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E27:9_2"/>
( <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font> or <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:9"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C</font></span>)</span> is   <a href="taxonom2.html#NM1" title="TAXONOM2:NM.1">Tree</a>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:9"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="taxonom2.html#E11:9"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="taxonom2.html#D2" target="_self" title="TAXONOM2:def.2">Def2</a></i>;<br/>


</div>
