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

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

<b>thus </b><a NAME="E1:23"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Olive">b<sub>1</sub></font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:23_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:23_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E4:23_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="partit1.html#E2:23_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:23_1_1"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:23_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="partit1.html#E4:23_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:23_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="partit1.html#E7:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="partit1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E9:23_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="partit1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:23_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="partit1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:23_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:23_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="partit1.html#E7:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="partit1.html#E11:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E3:23_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="partit1.html#E2:23_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="partit1.html#E2:23_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:23_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E5:23_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="partit1.html#E11:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="partit1.html#E2:23_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:23_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="partit1.html#E1:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="partit1.html#E2:23_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="partit1.html#E4:23_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:23_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="partit1.html#E11:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E14:23_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="partit1.html#E7:23_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#NR1" target="_self">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>


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


</div>
