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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">T</font> =  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a> ;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0;<br/>
<a NAME="E1:35"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:35_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:35_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:35_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0 <a href="hidden.html#R2">in</a>  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a> 
 ;<br/><a NAME="E4:35_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i>, </i>;<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="E2:35_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a> <b> such that </b><br/><a NAME="E4:35_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E5:35_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a> 
 
;<br/>
<a NAME="E6:35_1"/><b>then </b>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
 
;<br/>
<a NAME="E7:35_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35"/>
<a href="amistd_3.html#K1" target="_self">TrivialInfiniteTree</a>  <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/>


</div>
