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

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>;<br/>







<a NAME="E1:24"/>
 <a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:24"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E3:24"/><i><font color="Green">E23</font></i>: 
 <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </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="E1:24_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><a NAME="E2:24_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:24_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:24_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 </span><b>by </b><i><a class="txt" href="substlat.html#E2:24"><i><font color="Green">E19</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="24_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:24_1_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E5:24_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:24_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="substlat.html#E2"><i><font color="Green">Lemma20</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:24_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="substlat.html#E3:24"><i><font color="Green">E23</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:24_1_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="substlat.html#E2"><i><font color="Green">Lemma20</font></i></a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="substlat.html#E2"><i><font color="Green">Lemma20</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div><a NAME="E5:24_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:24_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24"/>
 <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </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="E1:24_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/><a NAME="E3:24_2"/>
for <font color="Olive">b</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font> &amp; <font color="Olive">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:24_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:24_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:24"/>
 <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">A</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
