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

<b>let </b><font color="Maroon">Y</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">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>, <font color="Maroon">PB</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">PA</font> <a href="partit1.html#NR2" target="_self">'&gt;'</a> <font color="Maroon">PB</font>
 ;<br/>

<a NAME="E2:9"/><i><font color="Green">E43</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PB</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> holds <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) ) ) )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">PA</font> <a href="setfam_1.html#R2">is_coarser_than</a> <font color="Maroon">PB</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:9"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> holds <br/><font color="Olive">b</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 ;<br/>

<b>set </b><font color="Maroon">B0</font> = <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">xb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">xb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D3">SETFAM_1:def 3</a></i>;<br/>
<a NAME="E4:9_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">xb</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:9_1"/>
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> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span>  holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </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><a NAME="E1:9_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:9_1_1"/><b>then </b>
 ex <font color="Olive">x8</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 
;<br/>
<b>hence </b><a NAME="E3:9_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:9_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">PB</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:9_1"/>
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">b</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </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><a NAME="E1:9_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:9_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y1</font> )
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>
<a NAME="E6:9_1_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">b</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">y1</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E7:9_1_2"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E8:9_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:9_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:9_1"/>
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>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span>  holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>proof </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><a NAME="E1:9_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:9_1_3"/>
 ex <font color="Olive">x8</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:9_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:9_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:9_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">x8</font> where <font color="Olive">x8</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">x8</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E12:9_1"/>
<font color="Maroon">b</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:9"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> holds <br/><font color="Olive">b</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font>
 ;<br/>


</div>
