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

<b>let </b><font color="Maroon">Z1</font> be   <a href="trees_1.html#NM1">Tree</a>, <font color="Maroon">Z2</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><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> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 ;<br/>

<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">Z1</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">Z2</font>;<br/>
<b>let </b><font color="Maroon">v</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">Z1</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">Z2</font>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">Z1</font>;<br/>



<b>assume </b><a NAME="E2:17"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; not <font color="Maroon">p</font>,<font color="Maroon">w</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 ;<br/>

<a NAME="E3:17"/><b>then </b><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">w</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">w</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </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"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font>
 ;<br/>

<a NAME="E2:17_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><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></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">Z2</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<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> <b> such that </b><br/><a NAME="E4:17_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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> &amp; <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">Z2</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:17_1_1"/>
<font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1_1"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 ;<br/>

<a NAME="E2:17_1_1_1"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t</font> )
 
<b>by </b><i>, , <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E3:17_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1"/>
( <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="trees_1.html#T32">TREES_1:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:17_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:17_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font>
 ;<br/><a NAME="E3:17_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><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></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</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="hidden.html#R2">in</a> <font color="Maroon">Z1</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<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> <b> such that </b><br/><a NAME="E5:17_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</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> &amp; <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</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="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 ;<br/><a NAME="E6:17_1"/>
not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</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>
 <b>by </b><i>, <a class="ref" href="trees_1.html#T32">TREES_1:32</a></i>;<br/><a NAME="E7:17_1"/><b>then </b>
<font color="Maroon">v</font> <a href="finseq_1.html#K8">^</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="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">Z2</font>
 <b>by </b><i>, , , <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><a NAME="E8:17_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">v</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">Z2</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:17"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
