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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">p</font> being <a href="trees_3.html#V5" target="_self">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="trees_3.html#K13" target="_self">tree</a> <font color="Olive">p</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/>
<a NAME="E1:126_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="trees_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:126_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:126_1_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">p</font> being <a href="trees_3.html#V5" target="_self">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> holds <br/> <a href="trees_3.html#K13" target="_self">tree</a> <font color="Olive">p</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be  <a href="trees_3.html#V5" target="_self">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E2:126_1_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:126_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="trees_3.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E4:126_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:126_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as  <a href="trees_3.html#V5" target="_self">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, </i>;<br/>
<a NAME="E8:126_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E9:126_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E10:126_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E11:126_1_1_1"/><b>then </b>
 <a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">q</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="trees_3.html#T17" target="_self">Th17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2">elementary_tree</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> <b>by </b><i><a class="ref" href="trees_1.html#T52">TREES_1:52</a></i>;<br/>
<a NAME="E13:126_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> is <a href="trees_3.html#V5" target="_self">FinTree-yielding</a>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> <b>by </b><i/>;<br/>
<a NAME="E15:126_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:126_1_1_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#T55">TREES_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E18:126_1_1_1"/>
 <a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">r</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="trees_3.html#D6" target="_self">Def6</a>, <a class="ref" href="trees_3.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E19:126_1_1_1"/>
 <a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">p</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:126_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="trees_3.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_3.html#T16" target="_self">Th16</a>);<br/></i>
<a NAME="E4:126_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> implies  <a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">p</font> is <a href="finset_1.html#V1">finite</a> )
 
;<br/>
<b>hence </b><a NAME="E5:126_1_1"/>
 <a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">p</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div>
