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

<b>let </b><font color="Maroon">P</font> be  <a href="finsub_1.html#V4" target="_self">preBoolean</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Q</font> be  <a href="finsub_1.html#V4" target="_self">preBoolean</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff ( <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">X</font> is <a href="finset_1.html#V1">finite</a> ) )
 <b>and </b><br/><a NAME="E2:17_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> iff ( <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">X</font> is <a href="finset_1.html#V1">finite</a> ) )
 ;<br/>

<a NAME="E3:17_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:17_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:17_1_2_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D4" target="_self">Def4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:17_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:17_1_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_1_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
