<?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">G</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">G</font> is <a href="bvfunc_2.html#V2">independent</a>
 ;<br/>

<b>let </b><font color="Maroon">P</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>, <font color="Maroon">Q</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:16"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:16_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E31</font></i>: <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6">%O</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font>
<b>by </b><i/>
;<br/><span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6">%O</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E5:16_1_1"/>
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="partit_2.html#R1" target="_self">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:16_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6">%O</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font>
<b>by </b><i/>
;<br/><span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6">%O</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <font color="Maroon">Y</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E5:16_1_2"/>
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="partit_2.html#R1" target="_self">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3"><b>suppose </b></a><a NAME="E1:16_1_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="partit_2.html#D1" target="_self">PARTIT_2:def 1</a><br/><b>assume </b><a NAME="E2:16_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:16_1_3"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span>
 <b>and </b><br/><a NAME="E5:16_1_3"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T51">RELSET_1:51</a></i>;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E7:16_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E8:16_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E9:16_1_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#D6">PARTIT1:def 6</a></i>;<br/><b>consider </b><font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E11:16_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E12:16_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E13:16_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="partit1.html#D6">PARTIT1:def 6</a></i>;<br/><b>consider </b><font color="Maroon">hP</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">FP</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E15:16_1_3"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">hP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E16:16_1_3"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">hP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FP</font>
 <b>and </b><br/><a NAME="E17:16_1_3"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">hP</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font>
 <b>and </b><br/><a NAME="E18:16_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FP</font>
 <b>and </b><br/><a NAME="E19:16_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="txt" href="partit_2.html#E1:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="bvfunc_2.html#D1">BVFUNC_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon">hQ</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">FQ</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E21:16_1_3"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">hQ</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E22:16_1_3"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">hQ</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FQ</font>
 <b>and </b><br/><a NAME="E23:16_1_3"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/><font color="Maroon">hQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font>
 <b>and </b><br/><a NAME="E24:16_1_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FQ</font>
 <b>and </b><br/><a NAME="E25:16_1_3"/>
<font color="Maroon">B</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="bvfunc_2.html#D1">BVFUNC_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font>, <font color="Maroon">Q</font> = <font color="Maroon">Q</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="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i/>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">P</font>) <b>-&gt; </b>   <a href="bvfunc_1.html#M1">Element</a> of <font color="Maroon">a<sub>1</sub></font> =  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">hP'</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">P</font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E28:16_1_3"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">p</font> being   <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">P</font> holds  <font color="Maroon">hP'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">p</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E29:16_1_3"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">P</font> ;<br/>
<a NAME="E3:16_1_3_1"/>
<font color="Maroon">hP'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:16_1_3_1"/>
<font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 ;<br/>


</div><b>end;</b></div><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">Q</font>) <b>-&gt; </b>   <a href="bvfunc_1.html#M1">Element</a> of <font color="Maroon">a<sub>1</sub></font> =  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">hQ'</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Q</font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E31:16_1_3"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">p</font> being   <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">Q</font> holds  <font color="Maroon">hQ'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">p</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E32:16_1_3"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/><font color="Maroon">hQ'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">Q</font> ;<br/>
<a NAME="E3:16_1_3_2"/>
<font color="Maroon">hQ'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="partit_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_3_2"/>
<font color="Maroon">hQ'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">FP'</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hP'</font>, <font color="Maroon">FQ'</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">hQ'</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/><b>set </b><font color="Maroon">A'</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FP'</font>;<br/><b>set </b><font color="Maroon">B'</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FQ'</font>;<br/><a NAME="E34:16_1_3"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">hP'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:16_1_3"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FP'</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FP'</font>
 ;<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="E3:16_1_3_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">hP'</font>
 <b>and </b><br/><a NAME="E4:16_1_3_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">P</font> <b>by </b><i/>;<br/>
<a NAME="E6:16_1_3_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:16_1_3_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:16_1_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FP'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><a NAME="E37:16_1_3"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">hQ'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E38:16_1_3"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FQ'</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FQ'</font>
 ;<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="E3:16_1_3_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">hQ'</font>
 <b>and </b><br/><a NAME="E4:16_1_3_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">hQ'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">Q</font> <b>by </b><i/>;<br/>
<a NAME="E6:16_1_3_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="partit_2.html#T3" target="_self">Th3</a>, <a class="ref" href="partit_2.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_3_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E39:16_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FQ'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</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="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i>, , <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E41:16_1_3"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">d</font> being   <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">G'</font>  holds  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">d</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">G'</font>;<br/>

<a NAME="E1:16_1_3_5"/>
<font color="Maroon">d</font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">d</font>
 
;<br/>
<b>hence </b><a NAME="E2:16_1_3_5"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">h'</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">G'</font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E43:16_1_3"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">d</font> being   <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">G'</font> holds  <font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">d</font>)
 <b>from </b><i><a class="ref" href="nattra_1.html#S1">NATTRA_1:sch 1</a>(<a class="ref" href="partit_2.html#T7" target="_self">Th7</a>);<br/></i><b>set </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font>;<br/><i><font color="Green">E90</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Q</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">P</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="partit_2.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/><a NAME="E45:16_1_3"/><i><font color="Green">E91</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">hQ'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/><a NAME="E46:16_1_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">h'</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">hP'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/><a NAME="E47:16_1_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E48:16_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">hQ'</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E49:16_1_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E1:16_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E51:16_1_3"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_7"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d'</font> = <font color="Maroon">d</font> as    <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">P</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_3_7_1"/>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> or  not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_7_1_1"><b>suppose </b></a><a NAME="E1:16_1_3_7_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:16_1_3_7_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">hQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E3:16"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E3:16_1_3_7_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">hP</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T1" target="_self">Th1</a>, <a class="txt" href="partit_2.html#E5:16_1_3"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E4:16_1_3_7_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">hP</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FP</font>
 <b>by </b><i>, , <a class="txt" href="partit_2.html#E5:16_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:16_1_3_7_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hP</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><a NAME="E6:16_1_3_7_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hP</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E2:16"><i><font color="Green">E19</font></i></a>, , <a class="ref" href="partit_2.html#T11" target="_self">Th11</a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><a NAME="E7:16_1_3_7_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">hQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FQ</font>
 <b>by </b><i>, , <a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:16_1_3_7_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><a NAME="E9:16_1_3_7_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="txt" href="partit_2.html#E17"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> = 
<font color="Maroon">hQ'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">d'</font>
<b>by </b><i><a class="ref" href="partit_2.html#T3" target="_self">Th3</a>, <a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<font color="Maroon">hP</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E9:16_1_3"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<font color="Maroon">d'</font>
<b>by </b><i><a class="txt" href="partit_2.html#E9:16_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="partit_2.html#T13" target="_self">Th13</a>, <a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>
<br/>.= 
<font color="Maroon">hQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="partit_2.html#E11:16_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">d'</font>
<b>by </b><i><a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>
<br/>.= 
<font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_7_1_2"><b>suppose </b></a><a NAME="E1:16_1_3_7_1_2"/>
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="partit_2.html#T5" target="_self">Th5</a>, <a class="txt" href="partit_2.html#E5:16_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E52:16_1_3"/>
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_8"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:16_1_3_8"/>
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>then </b><font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/>
<a NAME="E4:16_1_3_8"/><b>then </b><i><font color="Green">E105</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> or <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="txt" href="partit_2.html#E5:16_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_3_8_2"/>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> or <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_8_2_1"><b>suppose </b></a><a NAME="E1:16_1_3_8_2_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:16_1_3_8_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hQ'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:16_1_3_8_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T4" target="_self">Th4</a>, <a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_8_2_2"><b>suppose </b></a><a NAME="E1:16_1_3_8_2_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:16_1_3_8_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E4:16_1_3"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E3:16_1_3_8_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T2" target="_self">Th2</a>, <a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3_8_2_3"><b>suppose </b></a><a NAME="E1:16_1_3_8_2_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:16_1_3_8_2_3"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:16_1_3_8_2_3"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">hP'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/><a NAME="E4:16_1_3_8_2_3"/>
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:16_1_3_8_2_3"/><b>then </b>
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">hP'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span>
 <b>by </b><i><a class="ref" href="partit_2.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E6:16_1_3_8_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E7:16_1_3_8_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T8" target="_self">Th8</a>, <a class="txt" href="partit_2.html#E9:16_1_3"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E53:16_1_3"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</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="bvfunc_2.html#D5">BVFUNC_2:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E55:16_1_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E56:16_1_3"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FP'</font> holds <br/><font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_9"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_9"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FP'</font>
 ;<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="E3:16_1_3_9"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">hP'</font>
 <b>and </b><br/><a NAME="E4:16_1_3_9"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:16_1_3_9"/>
<font color="Maroon">hP'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E4:16_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E6:16_1_3_9"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a>, , <a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_3_9"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E5:16_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E57:16_1_3"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FP'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><a NAME="E58:16_1_3"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FQ'</font> holds <br/><font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_10"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_3_10"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FQ'</font>
 ;<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="E3:16_1_3_10"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">hQ'</font>
 <b>and </b><br/><a NAME="E4:16_1_3_10"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">hQ'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="partit_2.html#M1" target="_self">Element</a> of <font color="Maroon">Q</font> <b>by </b><i><a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E6:16_1_3_10"/>
<font color="Maroon">hQ'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hP'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">hQ'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<a NAME="E7:16_1_3_10"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , , <a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="partit_2.html#E9:16_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1_3_10"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E5:16_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E59:16_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FQ'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><a NAME="E60:16_1_3"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FQ'</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="bvfunc_2.html#D1">BVFUNC_2:def 1</a></i>;<br/><a NAME="E61:16_1_3"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="partit_2.html#E8:16_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="partit1.html#D6">PARTIT1:def 6</a></i>;<br/><a NAME="E62:16_1_3"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FP'</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T2" target="_self">Th2</a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a>, <a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_2.html#D1">BVFUNC_2:def 1</a></i>;<br/><a NAME="E63:16_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="partit_2.html#E7:16_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="partit1.html#D6">PARTIT1:def 6</a></i>;<br/><b>hence </b><a NAME="E64:16_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="partit_2.html#E9:16_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relset_1.html#T51">RELSET_1:51</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
