<?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  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">Pmax</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:36"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">hy</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">hy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> &amp; <font color="Olive">hy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> )
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> or <font color="Olive">x</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">x</font> or <font color="Maroon">h</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">x</font> )
 ;<br/>

<a NAME="E4:36"/><i><font color="Green">E35</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Pmax</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>let </b><font color="Maroon">Pb</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:36"/><i><font color="Green">E47</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="Olive">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">h</font> ) )
 ;<br/>

<b>set </b><font color="Maroon">P</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="E6:36"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Pb</font> <a href="tarski.html#R1">c=</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_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E49</font></i>: 
<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="E8:36"/><i><font color="Green">E51</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">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_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font> &amp; <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font> )
 ;<br/><b>consider </b><font color="Maroon">hy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E56</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> )
 <b>by </b><i/>;<br/><a NAME="E4:36_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">hy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:36_1"/>
<font color="Maroon">hy</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="taxonom2.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E6:36_1"/><b>then </b>
<font color="Maroon">hy</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:36"><i><font color="Green">E35</font></i></a>, </i>;<br/><a NAME="E7:36_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hy</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>hence </b><a NAME="E8:36_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:36"/><i><font color="Green">E58</font></i>: 
<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="36_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="E1:36_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/>

<b>thus </b><a NAME="E2:36_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E3:36"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:36"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">Pb</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:36"/>
<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>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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="E1:36_3"/>
<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/>

<a NAME="E2:36_3"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:36_3"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="taxonom2.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:36"/><b>then </b><i><font color="Green">E65</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>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E3:36"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E14:36"/><b>then </b><i><font color="Green">E66</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="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E15:36"/><i><font color="Green">E67</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_4"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</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="36_4_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="E1:36_4_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_4_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</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="taxonom2.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>thus </b><a NAME="E4:36_4_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:36"><i><font color="Green">E35</font></i></a>, </i>;<br/>


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

<b>thus </b><a NAME="E2:36_4"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_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="E1:36_4_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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="E3:36_4_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_4_2_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> or  not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_2_1_1"><b>suppose </b></a><a NAME="E1:36_4_2_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_4_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:36"><i><font color="Green">E29</font></i></a>, , <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="36_4_2_1_2"><b>suppose </b></a><a NAME="E1:36_4_2_1_2"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/><div class="add"><a NAME="E2:36_4_2_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">t</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">h</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_4_2_1_2_1"/>
( <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> or <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> )
 </span><b>by </b><i><a class="ref" href="taxonom2.html#T1" target="_self">Th1</a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:36_4_2_1_2_1_1"/>
<font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><a NAME="E2:36_4_2_1_2_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:36_4_2_1_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E2:36"><i><font color="Green">E31</font></i></a>, , <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="36_4_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:36_4_2_1_2_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font>
 ;<br/><div class="add"><a NAME="E2:36_4_2_1_2_1_2"/>
<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/><b>hence </b><a NAME="E3:36_4_2_1_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Pb</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="taxonom2.html#E2:36"><i><font color="Green">E31</font></i></a>, , , <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></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="36_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:36_5"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">A</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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_5_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> or <font color="Maroon">A</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> )
 </span><b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</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="36_5_1_1_1"><b>suppose </b></a><a NAME="E1:36_5_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/><div class="add"><a NAME="E2:36_5_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:36_5_1_1_1"/>
<font color="Maroon">A</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_5_1_1_2"><b>suppose </b></a><a NAME="E1:36_5_1_1_2"/>
<font color="Maroon">A</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/><div class="add"><b>hence </b><a NAME="E2:36_5_1_1_2"/>
<font color="Maroon">A</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:36_5"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>thus </b><a NAME="E4:36_5"/>
for <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">B</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> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:36_5_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">B</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>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_5_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> or <font color="Maroon">A</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> )
 </span><b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</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="36_5_2_1_1"><b>suppose </b></a><a NAME="E1:36_5_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_5_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_5_2_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> or <font color="Maroon">B</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> )
 </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="36_5_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_5_2_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_5_2_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_5_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_5_2_1_1_1_2"/>
<font color="Maroon">B</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/><div class="add"><a NAME="E2:36_5_2_1_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_5_2_1_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</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="36_5_2_1_2"><b>suppose </b></a><a NAME="E1:36_5_2_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">A</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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_5_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_5_2_1_2_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> or <font color="Maroon">B</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> )
 </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="36_5_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:36_5_2_1_2_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/><div class="add"><a NAME="E2:36_5_2_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_5_2_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_5_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:36_5_2_1_2_1_2"/>
<font color="Maroon">B</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/><div class="add"><a NAME="E2:36_5_2_1_2_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_5_2_1_2_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</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></div><b>end;</b></div>
<b>hence </b><a NAME="E17:36"/>
<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#T2" target="_self">Th2</a>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>

<b>thus </b><a NAME="E18:36"/>
<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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a><br/>

<b>assume </b><a NAME="E1:36_6"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_6_1"/>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> or  not <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1_1"><b>suppose </b></a><a NAME="E1:36_6_1_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_6_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</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> &amp; <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E2:36"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1_2"><b>suppose </b></a><a NAME="E1:36_6_1_2"/><i><font color="Green">E121</font></i>: 
not <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_6_1_2_1"/>
( <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> or <font color="Maroon">h</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">x</font> )
 </span><b>by </b><i><a class="ref" href="taxonom2.html#T1" target="_self">Th1</a>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1_2_1_1"><b>suppose </b></a><a NAME="E1:36_6_1_2_1_1"/>
<font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:36_6_1_2_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E3:36_6_1_2_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</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> &amp; <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 <b>by </b><i>, <a class="txt" href="taxonom2.html#E2:36"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1_2_1_2"><b>suppose </b></a><a NAME="E1:36_6_1_2_1_2"/>
<font color="Maroon">h</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:36_6_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 <b>by </b><i>, <a class="ref" href="taxonom2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E3:36_6_1_2_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</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> &amp; <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:36"><i><font color="Green">E29</font></i></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>

<b>let </b><font color="Maroon">Pmin</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E19:36"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">Pmax</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">Pmin</font>
 ;<br/>

<b>let </b><font color="Maroon">hw</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E20:36"/><i><font color="Green">E123</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> )
 ;<br/>

<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="setfam_1.html#D2">SETFAM_1:def 2</a><br/>

<b>assume </b><a NAME="E21:36"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">s</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>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_7"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font> or <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> )
 </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="36_7_1"><b>suppose </b></a><a NAME="E1:36_7_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pb</font>
 ;<br/><div class="add"><a NAME="E2:36_7_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmax</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:36_7_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmin</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T3" target="_self">Th3</a>, <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><b>thus </b><a NAME="E5:36_7_1"/>
 ex <font color="Olive">s1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmin</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">s1</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_2"><b>suppose </b></a><a NAME="E1:36_7_2"/>
<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/><div class="add"><a NAME="E2:36_7_2"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_7_2"/>
 ex <font color="Olive">s1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Pmin</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">s1</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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