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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="trees_9.html#V2">finite-branching</a> <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>;<br/>
<a NAME="E1:17"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E2:17"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:17_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T18" target="_self">Th18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ;<br/>
<a NAME="E3:17_1"/><i><font color="Green">E20</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E1:17_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:17_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E6:17_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:17_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="qc_lang4.html#E5:17_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E22</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="qc_lang4.html#E1:17_1_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<a NAME="E4:17_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:17_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:17_1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E4:17_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:17_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:17_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:17_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:17_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:17_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="qc_lang4.html#E3:17_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="qc_lang4.html#E4:17_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="qc_lang4.html#E6:17_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_1_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T28">CARD_1:28</a></i>;<br/>
<a NAME="E6:17_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:17_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="qc_lang4.html#E3:17_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/>
<b>hence </b><a NAME="E7:17_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E4:17_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:17_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:17_1_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:17_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:17_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:17_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="qc_lang4.html#E3:17_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:17"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="qc_lang4.html#E1:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="qc_lang4.html#E2:17"><i><font color="Green">E17</font></i></a>);</i><br/>


</div>
