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

<b>let </b><font color="Maroon">T</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">T1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">T1</font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">T</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:31"/>
not <font color="Maroon">f</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="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">f</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="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> </span>}</span> 
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t</font> )
 
;<br/>
<b>hence </b><a NAME="E3:31_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:31"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : verum </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
 ex <font color="Olive">t1</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> st <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t1</font>
 ;<br/>


</div>
