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<b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">T1</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>assume </b><a NAME="E1:3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:3"/>
 <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> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E29</font></i>: 
<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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="trees_1.html#T44">TREES_1:44</a></i>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">p</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="E1:3_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</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">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> )
 ;<br/><a NAME="E2:3_1_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="trees_1.html#D14">TREES_1:def 14</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p'</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:3_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">q</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E5:3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> </span>}</span> 
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> </span>}</span> 
 ;<br/>

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">p</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> </span>}</span> 
 ;<br/><a NAME="E2:3_2"/><b>then </b>
 ex <font color="Olive">p</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> ex <font color="Olive">s</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">s</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/><b>hence </b><a NAME="E3:3_2"/>
<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/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> : for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> </span>}</span> 
 ;<br/><a NAME="E2:3_3"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> not <font color="Olive">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t</font> ) )
 ;<br/><b>hence </b><a NAME="E3:3_3"/>
<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/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:3"/>
<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="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div>
