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

<b>set </b><font color="Maroon" title="c1">e</font> =  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2;<br/>
<a NAME="E1:21"/><i><font color="Green" title="E20">A1</font></i>: 
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="trees_1.html#T90" title="TREES_1:th.90">TREES_1:90</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">Z</font> be   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/><b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">Z</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:21"/>
<font color="Maroon" title="c2">Z</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c4">x2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">T1</font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span>;<br/>
<a NAME="E4:21"/><i><font color="Green" title="E22">A3</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><a NAME="E2:21_1_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="trees_1.html#T16" title="TREES_1:th.16">TREES_1:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">s</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>thus </b><a NAME="E1:21_2"/>
(  not <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> or ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1"/>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_1_1_1"/>
( ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 <b>by </b><i><a class="ref" href="trees_1.html#T36" title="TREES_1:th.36">TREES_1:36</a>, <a class="ref" href="trees_1.html#T39" title="TREES_1:th.39">TREES_1:39</a>, <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="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">w</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> , <font color="Maroon" title="c8">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:21_2_1_1_2"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">w</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T150" title="FINSEQ_2:th.150">FINSEQ_2:150</a></i>;<br/><a NAME="E4:21_2_1_1_2"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21_2_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/><a NAME="E5:21_2_1_1_2"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:21_2_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="trees_1.html#T45" title="TREES_1:th.45">TREES_1:45</a></i>;<br/><a NAME="E6:21_2_1_1_2"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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="E7:21_2_1_1_2"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" title="MODAL_1:func.1">Root</a> <font color="Maroon" title="c2">Z</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:21_2_1_1_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="trees_2.html#T14" title="TREES_2:th.14">TREES_2:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1_2_1_1"/>
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> or <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E3:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="modal_1.html#E7:21_2_1_1_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_2_1_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="modal_1.html#E1:21_2_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="modal_1.html#E3:21_2_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:21_2_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="modal_1.html#E3:21_2_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="modal_1.html#E1:21_2_1_1_2_1_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="modal_1.html#T5" target="_self" title="MODAL_1:th.5">Th5</a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_2_1_1_2"/><i><font color="Green" title="E27">A9</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:21_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="modal_1.html#E1:21_2_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="modal_1.html#E3:21_2_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:21_2_1_1_2_1_1_2"/>
( ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21_2_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="modal_1.html#E1:21_2_1_1_2_1_1_2"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:21_2_1_1_2"/>
( ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>assume </b><a NAME="E2:21_2"/><i><font color="Green" title="E23">A10</font></i>: 
( ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_2_1"/>
( ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E2:21_2"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1"><b>suppose </b></a><a NAME="E1:21_2_2_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span> &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_2_1_1_1_1"/>
( ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> ) or  ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> ) )
 </span><b>by </b><i><a class="txt" href="modal_1.html#E1:21"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="modal_1.html#E1:21_2_2_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_1_1_1"/>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 &amp; not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c6">s</font> )
 ;<br/><div class="add"><a NAME="E2:21_2_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> or <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="trees_1.html#T90" title="TREES_1:th.90">TREES_1:90</a></i>;<br/><b>hence </b><a NAME="E3:21_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_2_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21"><i><font color="Green" title="E21">A2</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_2_2_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_2"><b>suppose </b></a><a NAME="E1:21_2_2_1_2"/>
 ex <font color="Olive" title="b1">w</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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font> &amp; <font color="Maroon" title="c6">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">w</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_2_2_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21"><i><font color="Green" title="E21">A2</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:21_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21"/>
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c3">x1</font></span>)</span></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c4">x2</font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E4:21"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></i>;<br/>


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