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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>





<b>set </b><font color="Maroon" title="c4">e0</font> =  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 0;<br/>
<b>set </b><font color="Maroon" title="c5">e1</font> =  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1;<br/>
<b>set </b><font color="Maroon" title="c6">e2</font> =  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2;<br/>
<a NAME="E1:63"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E2:63"/><i><font color="Green" title="E47">A2</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="trees_1.html#T88" title="TREES_1:th.88">TREES_1:88</a></i>;<br/>
<a NAME="E3:63"/><i><font color="Green" title="E48">A3</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1 &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1 )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:63"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E5:63"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E6:63"/>
 <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:63"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a></i>;<br/>

<a NAME="E7:63"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="modal_1.html#E3:63"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:63"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E9:63"/>
 <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:63"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a></i>;<br/>

<b>set </b><font color="Maroon" title="c7">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">A</font>;<br/>
<a NAME="E10:63"/><i><font color="Green" title="E49">A4</font></i>: 
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 )
 
<b>by </b><i><a class="ref" href="trees_1.html#T55" title="TREES_1:th.55">TREES_1:55</a></i>;<br/>
<a NAME="E11:63"/><i><font color="Green" title="E50">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E12:63"/><b>then </b><i><font color="Green" title="E51">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E10:63"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E13:63"/>
not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="trees_1.html#T89" title="TREES_1:th.89">TREES_1:89</a></i>;<br/>
<a NAME="E14:63"/><b>then </b><i><font color="Green" title="E52">A7</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E10:63"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="modal_1.html#E11:63"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="modal_1.html#E12:63"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/>
<a NAME="E15:63"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">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" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_2.html#K9" title="TREES_2:func.9">--&gt;</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E16:63"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E14:63"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E17:63"/>
 <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:63"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a></i>;<br/>


</div>
