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

<b>let </b><font color="Maroon">X</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">H</font> be  <a href="abian.html#V2">covering</a> <a href="taxonom2.html#V5" target="_self">T_3</a>  <a href="taxonom2.html#M1" target="_self">Hierarchy</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">H</font> is <a href="taxonom2.html#V6" target="_self">lower-bounded</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E31</font></i>: 
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">C1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C1</font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">P1</font>, <font color="Olive">P2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">P1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C1</font> &amp; <font color="Olive">P2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C1</font> &amp; not <font color="Olive">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">P2</font> holds <br/><font color="Olive">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">P1</font> ) holds <br/> ex <font color="Olive">PX</font>, <font color="Olive">PY</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">PX</font> <a href="hidden.html#R2">in</a> <font color="Olive">C1</font> &amp; <font color="Olive">PY</font> <a href="hidden.html#R2">in</a> <font color="Olive">C1</font> &amp; ( for <font color="Olive">PZ</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">PZ</font> <a href="hidden.html#R2">in</a> <font color="Olive">C1</font> holds <br/>( <font color="Olive">PZ</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">PY</font> &amp; <font color="Olive">PX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">PZ</font> ) ) )
 ;<br/>

<a NAME="E4:37"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> is <a href="taxonom2.html#V3" target="_self">hierarchic</a>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( for <font color="Olive">P</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> ) &amp; ( for <font color="Olive">P1</font>, <font color="Olive">P2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Olive">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">P2</font> holds <br/><font color="Olive">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">P1</font> ) );<br/>
<b>consider </b><font color="Maroon">RL</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">L</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RL</font> iff ( <font color="Olive">L</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">L</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E7:37"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="abian.html#T4">ABIAN:4</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="taxonom2.html#E1:37"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">PX</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:37"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PX</font>
 <b>and </b><br/><a NAME="E13:37"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">PX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#D2" target="_self">Def2</a>, , <a class="txt" href="taxonom2.html#E1:37"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E14:37"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:37_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PX</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:37"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">P2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:37_2"/><b>then </b>
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PX</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PX</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:37_2"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E17:37"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">RL</font>
 
<b>by </b><i>, <a class="txt" href="taxonom2.html#E2:37"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:37_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">RL</font>
 <b>and </b><br/><a NAME="E2:37_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_1"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/><a NAME="E2:37_3_1"/><b>then </b>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:37_3_1"/>
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> or <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">Y</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>;<br/><b>take </b><font color="Maroon">Y</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>;<br/><a NAME="E4:37_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_3_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon">L4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L4</font> &amp; <font color="Maroon">L4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:37_3_2"/>
<font color="Maroon">L4</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:37_3_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_3"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>then consider </b><font color="Maroon">L3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3_3"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L3</font> &amp; <font color="Maroon">L3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>thus </b><a NAME="E4:37_3_3"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">P2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_4"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><b>then consider </b><font color="Maroon">L1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3_4"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">L1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">L2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_3_4"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_3_4_1"/>
( <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L2</font> or <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L1</font> )
 </span><b>by </b><i><a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_4_1_1"><b>suppose </b></a><a NAME="E1:37_3_4_1_1"/>
<font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_3_4_1_1"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_4_1_2"><b>suppose </b></a><a NAME="E1:37_3_4_1_2"/>
<font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_3_4_1_2"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:37_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RL</font>
 <b>by </b><i>, , <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a></i>;<br/><b>thus </b><a NAME="E8:37_3"/>
for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:37"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RL</font>
 <b>and </b><br/><a NAME="E21:37"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RL</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> holds <br/> not <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#T175">ORDERS_1:175</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<a NAME="E23:37"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">C</font> is    <a href="taxonom1.html#M1">Classification</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P1</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>, <font color="Maroon">P2</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:37_4"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>thus </b><a NAME="E3:37_4"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_5"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:37_5"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">PX</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E3:37_5"/>
contradiction
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E4:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:37"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_6"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 ;<br/>

<b>consider </b><font color="Maroon">P</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:37_6_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:37_6_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:37_6"/>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1"/>
(  not <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font> or <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1"/><i><font color="Green">E123</font></i>: 
not <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">PS</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:37_6_2_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font>
 <b>and </b><br/><a NAME="E4:37_6_2_1_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">PS</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#D2" target="_self">Def2</a>, , <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E5:37_6_2_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">h</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>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">hx</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">hx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Olive">hx</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">hx</font> );<br/><b>consider </b><font color="Maroon">Ca</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:37_6_2_1_1"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> iff ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E8:37_6_2_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">Ca</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 ;<br/>

<b>thus </b><a NAME="E2:37_6_2_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">hx</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">hx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">hx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">hx</font> );<br/><b>consider </b><font color="Maroon">Cb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:37_6_2_1_1"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> iff ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">p</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E11:37_6_2_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">Cb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font>
 ;<br/>

<b>thus </b><a NAME="E2:37_6_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E169</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_3"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><b>assume </b><a NAME="E2:37_6_2_1_1_3"/><i><font color="Green">E171</font></i>: 
for <font color="Olive">hv</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">hv</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> holds <br/><font color="Olive">hv</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">h</font>
 ;<br/><b>consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:37_6_2_1_1_3"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:37_6_2_1_1_3"/>
<font color="Maroon">p</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E48</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E6:37_6_2_1_1_3"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <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">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:37_6_2_1_1_3"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E9:37_6_2_1_1_3"/>
not <font color="Maroon">v</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E14:37"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E10:37_6_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E14:37"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E13:37_6_2_1_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ca</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Cb</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_6_2_1_1_4"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ca</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Cb</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_4_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>consider </b><font color="Maroon">hv</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_4_1"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">hv</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">hv</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E6:37"><i><font color="Green">E47</font></i></a>, <a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:37_6_2_1_1_4_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E5:37_6_2_1_1_4_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">hv</font>
 
<b>by </b><i>, <a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_4_1_1"/>
( <font color="Maroon">hv</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> or <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hv</font> )
 </span><b>by </b><i>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, , <a class="txt" href="taxonom2.html#E14:37"><i><font color="Green">E51</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_4_1_1_1"/>
<font color="Maroon">hv</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_4_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Cb</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_4_1_1_2"/>
<font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hv</font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_4_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Cb</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:37_6_2_1_1_4"/>
<font color="Maroon">Ca</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Cb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:37_6_2_1_1"/><b>then </b><i><font color="Green">E181</font></i>: 
<font color="Maroon">Ca</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E15:37_6_2_1_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">Cb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E16:37_6_2_1_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">Ca</font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E184</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">P2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_5"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> )
 ;<br/><a NAME="E2:37_6_2_1_1_5"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E3:37_6_2_1_1_5"/><b>then </b>
( <font color="Maroon">P1</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">P2</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_5"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, <a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E18:37_6_2_1_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">Cb</font> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E187</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">P2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_6"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> )
 ;<br/><a NAME="E2:37_6_2_1_1_6"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E3:37_6_2_1_1_6"/><b>then </b>
( <font color="Maroon">P1</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">P2</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_6"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, , <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1"/>
( <font color="Maroon">Cb</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Cb</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1"/>
<font color="Maroon">Cb</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">PX</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Pmax</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_1"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">PX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> &amp; <font color="Maroon">Pmax</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> )
 <b>and </b><br/><a NAME="E4:37_6_2_1_1_7_1_1"/><i><font color="Green">E190</font></i>: 
for <font color="Olive">PZ</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">PZ</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> holds <br/>( <font color="Olive">PZ</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmax</font> &amp; <font color="Maroon">PX</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">PZ</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T1" target="_self">Th1</a>, , </i>;<br/><a NAME="E5:37_6_2_1_1_7_1_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">Pmax</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_1"/>
<font color="Maroon">Pmax</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E7:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">Pmax</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">h</font>;<br/><b>consider </b><font color="Maroon">Pb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_6_2_1_1_7_1_1"/><i><font color="Green">E193</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><b>set </b><font color="Maroon">PS1</font> = <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E10:37_6_2_1_1_7_1_1"/>
<font color="Maroon">Pb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Pmax</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/>

<b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E11:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">Pb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:37_6_2_1_1_7_1_1"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> or <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> or <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> or <font color="Maroon">h</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">a</font> )
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, , </i>;<br/><a NAME="E13:37_6_2_1_1_7_1_1"/><i><font color="Green">E197</font></i>: 
 ex <font color="Olive">hx</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">hx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> &amp; <font color="Olive">hx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">hx</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E14:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, , , , </i>;<br/><a NAME="E15:37_6_2_1_1_7_1_1"/><i><font color="Green">E199</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:37_6_2_1_1_7_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</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>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:37_6_2_1_1_7_1_1"/><i><font color="Green">E201</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_3"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:37_6_2_1_1_7_1_1_3"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:37_6_2_1_1_7_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</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="E18:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E19:37_6_2_1_1_7_1_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>set </b><font color="Maroon">C1</font> = <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E20:37_6_2_1_1_7_1_1"/>
<font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></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="E21:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E22:37_6_2_1_1_7_1_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><div><i><font color="Green">E207</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P3</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_4"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_1_4_1"/>
( <font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_4_1_1"/>
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_4_1_1"/>
<font color="Maroon">P3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_4_1_2"/>
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_4_1_2"/>
<font color="Maroon">P3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E24:37_6_2_1_1_7_1_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">Pmax</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, , , , , </i>;<br/><div><i><font color="Green">E211</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P3</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_5"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_1_5_1"/>
( <font color="Maroon">Ca</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Ca</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_1"/>
<font color="Maroon">Ca</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_5_1_1"/><b>then </b>
<font color="Maroon">P3</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmax</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E17:37"><i><font color="Green">E56</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_5_1_1"/>
( <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P3</font> or <font color="Maroon">P3</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">Ca</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1"/>
( <font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> or <font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E48</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="37_6_2_1_1_7_1_1_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pmin</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">PY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">Pmin</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">PY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> )
 <b>and </b><br/><a NAME="E4:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E216</font></i>: 
for <font color="Olive">PZ</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">PZ</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> holds <br/>( <font color="Olive">PZ</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">PY</font> &amp; <font color="Maroon">Pmin</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">PZ</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T1" target="_self">Th1</a>, <a class="txt" href="taxonom2.html#E14:37"><i><font color="Green">E51</font></i></a>, , </i>;<br/><a NAME="E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon">Pmin</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">Pmin</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">hw</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><i><font color="Green">E218</font></i>: 
( <font color="Maroon">hw</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmin</font> &amp; <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hw</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">hw</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E9:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon">Pmax</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmin</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">Pmin</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">Pmax</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E2:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E220</font></i>: 
( <font color="Maroon">Pmin</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">Pmax</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">Pmin</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmax</font>
 ;<br/><b>consider </b><font color="Maroon">hx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E222</font></i>: 
( <font color="Maroon">hx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmin</font> &amp; <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hx</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">hx</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a></i>;<br/><b>consider </b><font color="Maroon">hy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">hy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> &amp; <font color="Maroon">hy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">hy</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">hy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hx</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#D5" target="_self">Def5</a>, <a class="txt" href="taxonom2.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">hz</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">hz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> &amp; <font color="Maroon">hx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hz</font> )
 <b>by </b><i>, <a class="ref" href="taxonom2.html#D5" target="_self">Def5</a>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><a NAME="E9:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">hy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hz</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/>
not <font color="Maroon">hy</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E11:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">hy</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">hz</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E12:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">hy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hz</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E13:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">hx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E14:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="taxonom2.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37_6_2_1_1_7_1_1_5_1_2_1_1_1_1"/>
<font color="Maroon">Pmax</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmin</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, <a class="ref" href="taxonom2.html#D4" target="_self">Def4</a>, , <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/><b>then </b><i><font color="Green">E228</font></i>: 
<font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmin</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, , , , , <a class="ref" href="taxonom2.html#D3" target="_self">Def3</a>, <a class="ref" href="taxonom2.html#T6" target="_self">Th6</a>, </i>;<br/><a NAME="E11:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
<font color="Maroon">Pmin</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P3</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E21:37"><i><font color="Green">E77</font></i></a></i>;<br/><b>hence </b><a NAME="E12:37_6_2_1_1_7_1_1_5_1_2_1_1_1"/>
( <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P3</font> or <font color="Maroon">P3</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="taxonom2.html#D4" target="_self">Def4</a>, <a class="ref" href="setfam_1.html#T23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_5_1_2_1_1_2"/>
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_5_1_2_1_1_2"/><b>then </b>
<font color="Maroon">P3</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmax</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E17:37"><i><font color="Green">E56</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_5_1_2_1_1_2"/>
( <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P3</font> or <font color="Maroon">P3</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_5_1_2"/>
( <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P3</font> or <font color="Maroon">P3</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">P2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_1_6"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_1_6_1"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_1_6_1_1_1"/>
( <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_1_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">P1</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E3:37_6_2_1_1_7_1_1_6_1_1_1_1"/>
<font color="Maroon">P2</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_7_1_1_6_1_1_1_1"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, , , <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_1_1_2"/>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_6_1_1_1_2"/><b>then </b>
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</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>hence </b><a NAME="E3:37_6_2_1_1_7_1_1_6_1_1_1_2"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_2"/>
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_1_6_1_2"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_1_6_1_2_1"/>
( <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_1_6_1_2_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_2_1_1"/>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_6_1_2_1_1"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_1_6_1_2_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_1_6_1_2_1_2"/>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_1_6_1_2_1_2"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E27:37_6_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">RL</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E28:37_6_2_1_1_7_1_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E29:37_6_2_1_1_7_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green">E35</font></i></a>, </i>;<br/><b>hence </b><a NAME="E30:37_6_2_1_1_7_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">Cb</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Pmin</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">PY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_6_2_1_1_7_1_2"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">Pmin</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">PY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> )
 <b>and </b><br/><a NAME="E4:37_6_2_1_1_7_1_2"/><i><font color="Green">E237</font></i>: 
for <font color="Olive">PZ</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">PZ</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> holds <br/>( <font color="Olive">PZ</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">PY</font> &amp; <font color="Maroon">Pmin</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Olive">PZ</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T1" target="_self">Th1</a>, , <a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E48</font></i></a>, </i>;<br/><a NAME="E5:37_6_2_1_1_7_1_2"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">Pmin</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a>, <a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E17:37"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E6:37_6_2_1_1_7_1_2"/>
<font color="Maroon">Pmin</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E13:37"><i><font color="Green">E49</font></i></a>, <a class="txt" href="taxonom2.html#E17:37"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E7:37_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E239</font></i>: 
<font color="Maroon">Pmin</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">hw</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_6_2_1_1_7_1_2"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">hw</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmin</font> &amp; <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hw</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">hw</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E17:37"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E10:37_6_2_1_1_7_1_2"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hw</font>;<br/><b>consider </b><font color="Maroon">PT</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:37_6_2_1_1_7_1_2"/><i><font color="Green">E242</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PT</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><b>set </b><font color="Maroon">PS1</font> = <font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/><a NAME="E13:37_6_2_1_1_7_1_2"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">PT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E14:37_6_2_1_1_7_1_2"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PT</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E15:37_6_2_1_1_7_1_2"/>
<font color="Maroon">PT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PT</font>
 ;<br/>

<b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E16:37_6_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon">PT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:37_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E245</font></i>: 
<font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E18:37_6_2_1_1_7_1_2"/><i><font color="Green">E246</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font> or <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">hw</font> or <font color="Maroon">hw</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> or <font color="Maroon">hw</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">a</font> )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E19:37_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E247</font></i>: 
<font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, , , , </i>;<br/><a NAME="E20:37_6_2_1_1_7_1_2"/><i><font color="Green">E248</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_2"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:37_6_2_1_1_7_1_2_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:37_6_2_1_1_7_1_2_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">C1</font> = <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E21:37_6_2_1_1_7_1_2"/><i><font color="Green">E250</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E22:37_6_2_1_1_7_1_2"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></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="E23:37_6_2_1_1_7_1_2"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmin</font>
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T4" target="_self">Th4</a>, , , , , </i>;<br/><a NAME="E24:37_6_2_1_1_7_1_2"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><div><i><font color="Green">E254</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P3</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_3"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_2_3_1"/>
( <font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_3_1_1"/>
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_3_1_1"/>
<font color="Maroon">P3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:37"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_3_1_2"/>
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_3_1_2"/>
<font color="Maroon">P3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E256</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P3</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_4"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:37_6_2_1_1_7_1_2_4"/>
<font color="Maroon">Pmin</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P3</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E9:37"><i><font color="Green">E48</font></i></a>, <a class="txt" href="taxonom2.html#E15:37"><i><font color="Green">E55</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_2_4"/>
( <font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P3</font> or <font color="Maroon">P3</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T23">SETFAM_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">P2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_6_2_1_1_7_1_2_5"/><i><font color="Green">E258</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_2_5_1"/>
( <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_2_5_1_1_1"/>
( <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_2_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_1_1_1"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_2_5_1_1_1_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">P1</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><a NAME="E3:37_6_2_1_1_7_1_2_5_1_1_1_1"/>
<font color="Maroon">P2</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E21:37"><i><font color="Green">E77</font></i></a>, <a class="ref" href="partit1.html#D3">PARTIT1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:37_6_2_1_1_7_1_2_5_1_1_1_1"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T2" target="_self">Th2</a>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a>, <a class="txt" href="taxonom2.html#E21:37"><i><font color="Green">E77</font></i></a>, <a class="ref" href="taxonom2.html#D3" target="_self">Def3</a>, <a class="ref" href="taxonom1.html#D1">TAXONOM1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_1_1_2"/>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_2_5_1_1_1_2"/><b>then </b>
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:37_6_2_1_1_7_1_2_5_1_1_1_2"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_2"/>
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:37_6_2_1_1_7_1_2_5_1_2"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_6_2_1_1_7_1_2_5_1_2_1"/>
( <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <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="37_6_2_1_1_7_1_2_5_1_2_1_1"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_2_1_1"/>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_5_1_2_1_1"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_1_7_1_2_5_1_2_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_1_7_1_2_5_1_2_1_2"/>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_1_7_1_2_5_1_2_1_2"/>
( <font color="Maroon">P1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P2</font> or <font color="Maroon">P2</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E20:37"><i><font color="Green">E76</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E28:37_6_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E263</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">RL</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E29:37_6_2_1_1_7_1_2"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E30:37_6_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">PT</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Pmin</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">hw</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:37"><i><font color="Green">E35</font></i></a>, </i>;<br/><b>hence </b><a NAME="E31:37_6_2_1_1_7_1_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:37_6_2_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_6_2_1_2"><b>suppose </b></a><a NAME="E1:37_6_2_1_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_6_2_1_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E26:37"/>
( <font color="Maroon">C</font> is    <a href="taxonom1.html#M1">Classification</a> of <font color="Maroon">X</font> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T2" target="_self">Th2</a></i>;<br/>


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