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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">x'</font> where <font color="Olive">x'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x'</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E3:65_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:65_1_1_1"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </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">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:65_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:65_1_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:65_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:65_1_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:65_1_1_1_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</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="65_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:65_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:65_1_1_1_2_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:65_1_1_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>


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

<b>let </b><font color="Maroon">a</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="E2:65_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E4:65_1_1_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:65_1_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1_1_2_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:65_1_1_1_2_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:65_1_1_1_2_2_1"/><b>then </b>
not <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </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="E3:65_1_1_1_2_2_1"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:65_1_1_1_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <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="65_1_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:65_1_1_1_2_2_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><a NAME="E2:65_1_1_1_2_2_2"/><b>then </b>
not <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </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="E3:65_1_1_1_2_2_2"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:65_1_1_1_2_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <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>
<a NAME="E3:65_1_1_1"/>
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> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> 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> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> 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> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_3"><b>proof </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">Y</font>;<br/>

<b>assume </b><a NAME="E1:65_1_1_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:65_1_1_1_3"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> &amp; not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:65_1_1_1_3"/>
<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/>

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

<b>assume </b><a NAME="E4:65_1_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E5:65_1_1_1_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_3_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:65_1_1_1_3_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1_1_1_3_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:65_1_1_1_3_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1_1_1_3_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:65_1_1_1_3_1_1_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><b>thus </b><a NAME="E4:65_1_1_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:65_1_1_1_3_1_1_1"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:65_1_1_1_3_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:65_1_1_1_3_1"/>
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font>
 <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="65_1_1_1_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1_1_3_2"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_3_2_1"><b>suppose </b></a><a NAME="E1:65_1_1_1_3_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:65_1_1_1_3_2_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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_3_2_2"><b>suppose </b></a><a NAME="E1:65_1_1_1_3_2_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:65_1_1_1_3_2_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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_3_2_3"><b>suppose </b></a><a NAME="E1:65_1_1_1_3_2_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:65_1_1_1_3_2_3"/>
( <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/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_3_2_4"><b>suppose </b></a><a NAME="E1:65_1_1_1_3_2_4"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:65_1_1_1_3_2_4"/>
( <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/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_1_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</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>
<b>hence </b><a NAME="E4:65_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  </span>}</span> ,<span class="p2">{<span class="default"> <font color="Olive">x'</font> where <font color="Olive">x'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x'</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  </span>}</span> </span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>
 ;<br/>


</div>
