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

<b>let </b><font color="Maroon">f</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="funct_1.html#NM1">Function</a>;<br/>

<b>set </b><font color="Maroon">X</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>;<br/>
<b>let </b><font color="Maroon">P</font> be    <a href="eqrel_1.html#M1">a_partition</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>;<br/>

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> : verum </span>}</span> ;<br/>
<a NAME="E1:22"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">p</font>
 ;<br/>
<a NAME="E4:22_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> : verum </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">f</font> ;<br/>
<a NAME="E3:22"/>
<font color="Maroon">Q</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">f</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="22_2_1_1"><b>case </b></a><a NAME="E1:22_2_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:22_2_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/><b>thus </b><a NAME="E3:22_2_1_1"/>
<font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_2_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_2_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:22_2_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">P</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:22_2_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y</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">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:22_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">p</font> &amp; <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E8:22_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">f</font>;<br/><b>assume </b><a NAME="E4:22_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E6:22_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">p</font>
 ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">p</font>;<br/><a NAME="E9:22_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E10:22_2_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="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">f</font>;<br/><b>assume </b><a NAME="E11:22_2_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><b>then consider </b><font color="Maroon">p1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E13:22_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">p1</font>
 ;<br/><b>assume </b><a NAME="E14:22_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E15:22_2_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>assume </b><a NAME="E16:22_2_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:22_2_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:22_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E21:22_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> )
 <b>by </b><i>, , , , <a class="ref" href="relat_1.html#D11">RELAT_1:def 11</a></i>;<br/><b>hence </b><a NAME="E22:22_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_2"><b>case </b></a><a NAME="E1:22_2_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_2_1_2"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> : verum </span>}</span>  is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">f</font>
 ;<br/>


</div>
