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

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">T</font> where <font color="Olive">T</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : <font color="Olive">T</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> </span>}</span> ;<br/>
<a NAME="E1:4_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">T</font> where <font color="Olive">T</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : <font color="Olive">T</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="trees_3.html#K1" target="_self">Trees</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">T</font> where <font color="Olive">T</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : <font color="Olive">T</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> </span>}</span> 
 ;<br/>

<a NAME="E2:4_1_1_1"/><b>then </b>
 ex <font color="Olive">T</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> &amp; <font color="Olive">T</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> )
 
;<br/>
<b>hence </b><a NAME="E3:4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K1" target="_self">Trees</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">T</font> where <font color="Olive">T</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : <font color="Olive">T</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="trees_3.html#K1" target="_self">Trees</a>  ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

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

<b>thus </b><a NAME="E3:4_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <font color="Maroon">x</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:4_1_1_2"/><b>then </b>
 ex <font color="Olive">T</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> &amp; <font color="Olive">T</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> )
 
;<br/>
<b>hence </b><a NAME="E3:4_1_1_2"/>
<font color="Maroon">x</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>
 ;<br/>


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

<b>assume </b><a NAME="E4:4_1_1"/>
<font color="Maroon">x</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">x</font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/>
<a NAME="E6:4_1_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K1" target="_self">Trees</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:4_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div>
