<?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">e1</font> =  <a href="trees_1.html#K2">elementary_tree</a> 1;<br/>
<b>set </b><font color="Maroon">e2</font> =  <a href="trees_1.html#K2">elementary_tree</a> 2;<br/>
<a NAME="E1:73"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="modal_1.html#K17" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>set </b><font color="Maroon">E</font> = <span class="p1">(<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="p1"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">(<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="p1"><a href="domain_1.html#K1">[</a><span class="default">1,1</span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E2:73"/><i><font color="Green">E38</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> 1
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:73"/><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"><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">1,0</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> 1 &amp;  <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> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 1 )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:73"/><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"><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">1,0</span><a href="domain_1.html#K1">]</a></span></span>)</span> &amp; <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="E5:73"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <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> 1</span>)</span> <a href="trees_2.html#K9">--&gt;</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">1,0</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="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E6:73"/><i><font color="Green">E61</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">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> 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">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E7:73"/><i><font color="Green">E62</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"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="modal_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E8:73"/><i><font color="Green">E63</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"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="modal_1.html#T16" target="_self">Th16</a>, <a class="txt" href="modal_1.html#E6:73"><i><font color="Green">E61</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<b>thus </b><a NAME="E9:73"/>
 <a href="modal_1.html#K17" target="_self">VERUM</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="modal_1.html#E7:73"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>

<b>thus </b><a NAME="E10:73"/>
 <a href="modal_1.html#K17" target="_self">VERUM</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="modal_1.html#E8:73"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<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="E11:73"/><i><font color="Green">E64</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="E12:73"/><i><font color="Green">E69</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="E13:73"/><b>then </b><i><font color="Green">E70</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="txt" href="modal_1.html#E11:73"><i><font color="Green">E64</font></i></a>, <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E14:73"/>
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="E15:73"/><b>then </b><i><font color="Green">E71</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="txt" href="modal_1.html#E11:73"><i><font color="Green">E64</font></i></a>, <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, , <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E16:73"/><b>then </b>
 <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="E17:73"/><b>then </b><i><font color="Green">E76</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"><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="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<b>assume </b><a NAME="E18:73"/>
not  <a href="modal_1.html#K17" target="_self">VERUM</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E19:73"/><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="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a></i>;<br/>
<b>hence </b><a NAME="E20:73"/>
contradiction
 ;<br/>


</div>
