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<div class="add">

<b>let </b><font color="Maroon">T1</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="trees_1.html#M4">AntiChain_of_Prefixes</a> of <font color="Maroon">T</font>;<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="E1:9"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:9"/>
 ex <font color="Olive">q</font>, <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">q</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:9_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:9_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/>
<a NAME="E5:9_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E6:9_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E7:9_1"/>
 ex <font color="Olive">q</font>, <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a><br/>

<b>thus </b><a NAME="E4:9"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 ;<br/>

<a NAME="E2:9_2"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</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>

<b>thus </b><a NAME="E5:9"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:9_3"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_3_1"/>
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">r</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font> ) )
 ;<br/><a NAME="E2:9_3_1"/><b>then </b><i><font color="Green">E40</font></i>: 
not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:9_3_1"/>
<font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><b>then </b><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font> = 
<font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><a NAME="E5:9_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>hence </b><a NAME="E6:9_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</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="hs2(this)" href="javascript:()" title="9_3_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:9_3_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_3_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:9_3_2_1"/><b>then </b>
not <font color="Maroon">s</font>,<font color="Maroon">p</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, , <a class="ref" href="trees_1.html#D13">TREES_1:def 13</a></i>;<br/><b>hence </b><a NAME="E3:9_3_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:9_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:9_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>


</div>
