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

<b>let </b><font color="Maroon">W</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:25"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">w</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font>  holds not <font color="Olive">w</font> <a href="finseq_1.html#K8">^</a> <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> <font color="Maroon">W</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D2" target="_self">TREES_2:def 2</a><br/>

<a NAME="E2:25"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">w</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">w</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E3:25"/><i><font color="Green">E58</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<a NAME="E4:25"/><i><font color="Green">E60</font></i>: 
 <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">w</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="E1:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">w</font>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
<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="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/>
<a NAME="E5:25_1"/>
not <font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <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> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:25_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D5">TREES_1:def 5</a></i>;<br/>
<a NAME="E7:25_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:25_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E9:25_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="Maroon">w</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> =  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">w</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E6:25"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">w</font>
 ;<br/>

<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<a NAME="E7:25"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E9:25"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> </span>}</span> 
 
;<br/>
<a NAME="E10:25"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>from </b><i><a class="ref" href="trees_2.html#S3" target="_self">TREES_2:sch 3</a>();<br/></i>
<a NAME="E11:25"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T80">CARD_1:80</a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>hence </b><a NAME="E12:25"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
