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

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a><br/>

<b>thus </b><a NAME="E1:118"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13" target="_self">tree</a> <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:118_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13" target="_self">tree</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:118_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <a href="xboole_0.html#K1">{}</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> ) )
 
<b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a>, </i>;<br/>
<b>assume </b><a NAME="E3:118_1"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:118_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="xboole_0.html#K1">{}</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="trees_3.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<a NAME="E6:118_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="ref" href="trees_3.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E7:118_1"/>
contradiction
 ;<br/>


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

<b>assume </b><a NAME="E2:118"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 ;<br/>

<a NAME="E3:118"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<b>hence </b><a NAME="E4:118"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13" target="_self">tree</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a>, </i>;<br/>


</div>
