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

<b>let </b><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T</font> be   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/><b>let </b><font color="Maroon" title="c3">P</font> be    <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c2">T</font>;<br/><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<a NAME="E2:9"/>
 ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">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" title="c5">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:9_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:9_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>
<a NAME="E5:9_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font>
 
<b>by </b><i><a class="txt" href="trees_a.html#E4:9_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E6:9_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="txt" href="trees_a.html#E2:9_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="trees_a.html#E4:9_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E7:9_1"/>
 ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="trees_a.html#E2:9_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="trees_a.html#E5:9_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c1">T1</font>
 
<b>by </b><i><a class="ref" href="trees_a.html#D1" target="_self" title="TREES_A:def.1">Def1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1" title="TREES_2:def.1">TREES_2:def 1</a><br/>

<b>thus </b><a NAME="E4:9"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c1">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">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" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font>
 ;<br/>

<a NAME="E2:9_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c1">T1</font>
 
<b>by </b><i><a class="txt" href="trees_a.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="trees_a.html#D1" target="_self" title="TREES_A:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:9_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c1">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E3:9"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></i>;<br/>


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

<b>thus </b><a NAME="E5:9"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c1">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c1">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">p</font>
 ;<br/>

<a NAME="E2:9_3"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c1">T1</font>
 
<b>by </b><i><a class="txt" href="trees_a.html#E3:9"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></i>;<br/>
<div><i><font color="Green" title="E12">A7</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" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font> ) )
 ;<br/><a NAME="E2:9_3_1"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
not <font color="Maroon" title="c4">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><a NAME="E3:9_3_1"/>
<font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/><b>then </b><font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c4">p</font>
<b>by </b><i><a class="txt" href="trees_a.html#E2:9_3_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
;<br/><a NAME="E5:9_3_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/><b>hence </b><a NAME="E6:9_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">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" title="c6">s</font>, <font color="Maroon" title="c7">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:9_3_2"/><i><font color="Green" title="E13">A9</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">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" title="c6">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 ;<br/><a NAME="E2:9_3_2_1"/><b>then </b>
not <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c4">p</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="txt" href="trees_a.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="trees_a.html#E2:9_3_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="trees_1.html#D13" title="TREES_1:def.13">TREES_1:def 13</a></i>;<br/><b>hence </b><a NAME="E3:9_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_a.html#E2:9_3_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="trees_a.html#T1" target="_self" title="TREES_A:th.1">Th1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:9_3_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E2:9_3_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:9_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T1</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="trees_a.html#E2:9_3"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="trees_a.html#E3:9_3"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="trees_a.html#D1" target="_self" title="TREES_A:def.1">Def1</a></i>;<br/>


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


</div>
