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<b>let </b><font color="Maroon">T</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon">T1</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="trees_1.html#M4">AntiChain_of_Prefixes</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> &amp; <font color="Maroon">q</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  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Olive">s</font> is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1"/>
( 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="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> ) or  ex <font color="Olive">p</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">p</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>suppose </b></a><a NAME="E1:18_1_1"/><i><font color="Green">E29</font></i>: 
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="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">p'</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Maroon">r</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E33</font></i>: 
( <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> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_1"/>
<span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:18_1_1_1"/>
not <font color="Maroon">p'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:18_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:18_1_1"/>
 ex <font color="Olive">p'</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> ex <font color="Olive">r</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2"><b>suppose </b></a><a NAME="E1:18_1_2"/>
 ex <font color="Olive">p</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">p</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> )
 ;<br/><div class="add"><b>then consider </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><br/><a NAME="E3:18_1_2"/><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>  <a href="relat_1.html#K1">dom</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> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> )
 ;<br/><b>consider </b><font color="Maroon">p'</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>, <font color="Maroon">r'</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E5:18_1_2"/><i><font color="Green">E37</font></i>: 
( <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> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p'</font>
 ;<br/><a NAME="E2:18_1_2_1"/>
<font color="Maroon">p</font>,<font color="Maroon">p'</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:18_1_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="trees_1.html#D13">TREES_1:def 13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:18_1_2"/>
 ex <font color="Olive">p'</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> ex <font color="Olive">r</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">T</font>,<font color="Maroon">P</font>,<font color="Maroon">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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