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

<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">PA</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<a NAME="E1:24"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T19" target="_self" title="PARTIT1:th.19">Th19</a></i>;<br/>
<a NAME="E2:24"/>
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c2">PA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font>
 ;<br/>

<a NAME="E2:24_1_1"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="ref" href="partit1.html#D5" target="_self" title="PARTIT1:def.5">Def5</a></i>;<br/>
<a NAME="E3:24_1_1"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> ) )
 
<b>by </b><i><a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:24_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></i>;<br/>
<a NAME="E6:24_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="partit1.html#E1:24_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">a</font>;<br/>
<a NAME="E8:24_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E6:24_1_1"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E9:24_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E1:24_1_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E10:24_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:24_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E13:24_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c7">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:24_1_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E5:24_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="partit1.html#E6:24_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="partit1.html#E12:24_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="partit1.html#E2:24_1_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:24_1_1_1"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
not <font color="Maroon" title="c6">b</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">u</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E5:24_1_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:24_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="partit1.html#E2:24_1_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="partit1.html#E12:24_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="partit1.html#E2:24_1_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="partit1.html#E4:24_1_1_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:24_1_1"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:24_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:24_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E12:24_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="partit1.html#T8" target="_self" title="PARTIT1:th.8">Th8</a></i>;<br/>
<a NAME="E16:24_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E2:24_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="partit1.html#E14:24_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E17:24_1_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E12:24_1_1"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1"/>
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c2">PA</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c2">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">PA</font>
 <b>by </b><i><a class="txt" href="partit1.html#E1:24"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="partit1.html#T5" target="_self" title="PARTIT1:th.5">Th5</a></i>;<br/>


</div>
