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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="trees_1.html#K2">elementary_tree</a> 1;<br/>
<a NAME="E1:63"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> 1
 
<b>by </b><i><a class="ref" href="modal_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E2:63"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,1</span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:63"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">1,1</span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E5:63"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E1:63_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green">E46</font></i></a>, <a class="txt" href="modal_1.html#E3:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><b>assume </b><a NAME="E2:63_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_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="E1:63_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:63_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T33">TREES_1:33</a></i>;<br/><a NAME="E3:63_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green">E46</font></i></a>, <a class="txt" href="modal_1.html#E3:63"><i><font color="Green">E47</font></i></a>, <a class="txt" href="modal_1.html#E2:63_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:63_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:63_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:63"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K1" target="_self">Root</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E7:63"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:63"><i><font color="Green">E48</font></i></a>, <a class="ref" href="modal_1.html#T26" target="_self">Th26</a></i>;<br/>


</div>
