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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="taxonom1.html#M2">Strong_Classification</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:9"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:9"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a>, <a class="ref" href="taxonom1.html#D2">TAXONOM1:def 2</a></i>;<br/>
<a NAME="E4:9"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="taxonom2.html#E2:9"><i><font color="Green">E7</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_1.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_1.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="relset_1.html#NM3">Relation</a> of  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E6:9"/><i><font color="Green">E9</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #);<br/>
<a NAME="E7:9"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="taxonom1.html#M1">Classification</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a>, <a class="ref" href="taxonom1.html#D2">TAXONOM1:def 2</a></i>;<br/>
<a NAME="E8:9"/><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a>, <a class="ref" href="taxonom1.html#D2">TAXONOM1:def 2</a></i>;<br/>
<a NAME="E9:9"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="taxonom2.html#E8:9"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:9"/><i><font color="Green">E13</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="taxonom2.html#V1" target="_self">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">c<sub>8</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) <b>by </b><i><a class="txt" href="taxonom2.html#E8:9"><i><font color="Green">E11</font></i></a>, <a class="txt" href="taxonom2.html#E9:9"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="taxonom1.html#M2">Strong_Classification</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E5:9_1"/>
 <a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#D2">TAXONOM1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) <b>by </b><i><a class="txt" href="taxonom2.html#E3:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom2.html#D1" target="_self">TAXONOM2:def 1</a><br/>

<a NAME="E7:9_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E4:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="taxonom2.html#E3:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E8:9_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">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">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:9_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#D6">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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E2:9_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D2">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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><a NAME="E3:9_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1">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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><a NAME="E3:9_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E7:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1">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">c<sub>8</sub></font> <a href="orders_1.html#R8">is_superior_of</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #)
 <b>by </b><i><a class="txt" href="taxonom2.html#E8:9_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="orders_1.html#D13">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">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #);<br/><b>given </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #)<b> such that </b><a NAME="E1:9_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:9_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="taxonom2.html#E7:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a>, <a class="txt" href="taxonom2.html#E4:9_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E7:9_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:9_2"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E8:9_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:9_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E11:9_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E12:9_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:9"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E13:9_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E14:9_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E15:9_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:9"><i><font color="Green">E8</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E16:9_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green">E16</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:9_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a>, <a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:9_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a>, <a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E24:9_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="taxonom1.html#M1">Classification</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:9"><i><font color="Green">E6</font></i></a>, <a class="ref" href="taxonom1.html#D2">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">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="taxonom2.html#E24:9_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="taxonom1.html#D1">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">c<sub>19</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_2_1_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><b>thus </b><a NAME="E4:9_2_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="taxonom2.html#E13:9_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="taxonom2.html#E15:9_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="taxonom2.html#E3:9_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="taxonom2.html#T3" target="_self">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">c<sub>21</sub></font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_2_1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E22:9_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><b>thus </b><a NAME="E4:9_2_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E10:9_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="taxonom2.html#E12:9_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="taxonom2.html#E16:9_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="taxonom2.html#E19:9_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="taxonom2.html#E3:9_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="taxonom2.html#T3" target="_self">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">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E27:9_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9">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">InclPoset</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is   <a href="taxonom2.html#NM1">Tree</a>
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:9"><i><font color="Green">E9</font></i></a>, <a class="txt" href="taxonom2.html#E11:9"><i><font color="Green">E13</font></i></a>, <a class="ref" href="taxonom2.html#D2" target="_self">Def2</a></i>;<br/>


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