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

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



<b>set </b><font color="Maroon">e</font> =  <a href="trees_1.html#K2">elementary_tree</a> 2;<br/>
<b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="trees_2.html#K8">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>,<font color="Maroon">A</font>;<br/>
<a NAME="E1:64"/><i><font color="Green">E37</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="trees_1.html#K2">elementary_tree</a> 2 &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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> 2 )
 
<b>by </b><i><a class="ref" href="trees_1.html#T55">TREES_1:55</a></i>;<br/>
<a NAME="E2:64"/><i><font color="Green">E38</font></i>: 
 <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> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 2
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:64"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">A</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> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">2,0</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">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E4:64"/><b>then </b><i><font color="Green">E41</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"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="modal_1.html#T6" target="_self">Th6</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E5:64"/>
not <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="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="modal_1.html#T6" target="_self">Th6</a>, <a class="ref" href="modal_1.html#T7" target="_self">Th7</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E7:64"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</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"><span class="p4">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">A</font></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">1</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">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E8:64"/>
not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</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>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">o</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> <b>by </b><i><a class="ref" href="modal_1.html#T9" target="_self">Th9</a>, <a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E10:64"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E1:64_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> implies <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:64_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <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"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="modal_1.html#T6" target="_self">Th6</a>, <a class="ref" href="modal_1.html#T7" target="_self">Th7</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E3:64_1_1_1"/>
not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:64_1_1_1"/>
<font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="modal_1.html#T13" target="_self">Th13</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:64_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_2_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">s</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="64_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</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="64_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">w</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:64_1_1_2_1_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font> )
 ;<br/><a NAME="E2:64_1_1_2_1_2_1"/>
<font color="Maroon">o</font> <a href="tarski.html#R1">is_a_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:64_1_1_2_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <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"><span class="p3">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">A</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="modal_1.html#T13" target="_self">Th13</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><a NAME="E4:64_1_1_2_1_2"/>
<font color="Maroon">o</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="modal_1.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_1.html#T33">TREES_1:33</a></i>;<br/><a NAME="E5:64_1_1_2_1_2"/><b>then </b>
 ex <font color="Olive">w</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">w</font> )
 <b>by </b><i><a class="ref" href="modal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="modal_1.html#T6" target="_self">Th6</a>, <a class="ref" href="modal_1.html#T7" target="_self">Th7</a>, , <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><b>hence </b><a NAME="E6:64_1_1_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</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="E4:64_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:64_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">o</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="E11:64"/>
<font color="Maroon">o</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E12:64"/>
 <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">A</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="modal_1.html#T12" target="_self">Th12</a>, </i>;<br/>

<b>reconsider </b><font color="Maroon">u</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> <b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E14:64"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E1:64_2_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> implies <font color="Maroon">u</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><b>assume </b><a NAME="E2:64_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">u</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2_1_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">s</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="64_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_2_1_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_2_1_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</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="64_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_2_1_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:64_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">u</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T33">TREES_1:33</a></i>;<br/><a NAME="E3:64_2_1_1_1_2"/><b>then </b>
 ex <font color="Olive">w</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">u</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">w</font> )
 <b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="modal_1.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:64_2_1_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</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="E4:64_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:64_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">u</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="E15:64"/>
<font color="Maroon">u</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E16:64"/>
 <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">B</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"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="modal_1.html#T13" target="_self">Th13</a>, </i>;<br/>


</div>
