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

<a NAME="E1:81_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
;<br/>
<a NAME="E2:81_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive">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="orders_1.html#T4">ORDERS_1:4</a></i>;<br/>
<b>consider </b><font color="Maroon">Choice</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:81_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font> <a href="hidden.html#R1">=</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E5:81_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font>
 <b>by </b><i>, <a class="ref" href="wellord2.html#T28">WELLORD2:28</a></i>;<br/>
<a NAME="E6:81_1"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>)</span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="modelc_1.html#K1" target="_self">k_id</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="modelc_1.html#K1" target="_self">k_id</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>)</span>)</span>;<br/>
<a NAME="E1:81_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:81_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E2:81_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="modelc_1.html#K1" target="_self">k_id</a> <font color="Maroon">x</font>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D13" target="_self">Def13</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="modelc_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>);<br/>
<a NAME="E4:81_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>) is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="modelc_1.html#E11:81_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E5:81_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="orders_1.html#T5">ORDERS_1:5</a></i>;<br/>
<a NAME="E6:81_1_1_1"/><b>then </b>
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>)</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">x</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E7:81_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="modelc_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:81_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E9:81_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>2</sub></font>() <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:81_1_1_1"/><b>then </b>
<font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:81_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:81_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>();<br/>
<a NAME="E4:81_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">m</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">m</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:81_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">m</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">m</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="ref" href="modelc_1.html#T1" target="_self">Th1</a>);<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<i><font color="Green">E61</font></i>: <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 0 = 
<font color="Maroon">H<sub>2</sub></font>(0)
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K31" target="_self">|**</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="ref" href="funct_7.html#T86">FUNCT_7:86</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<a NAME="E8:81_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:81_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K31" target="_self">|**</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:81_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="modelc_1.html#K1" target="_self">k_id</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1)
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K31" target="_self">|**</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K31" target="_self">|**</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>)</span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D13" target="_self">Def13</a>, </i>
;<br/>
<b>hence </b><a NAME="E4:81_1_1_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>)</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:81_1_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>)</span>)</span> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E8:81_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>and </b><br/><a NAME="E9:81_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>)</span>)</span>
 ;<br/>
<a NAME="E10:81_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">s</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>);<br/>
<a NAME="E1:81_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>) is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="modelc_1.html#E11:81_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E2:81_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>) <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="orders_1.html#T5">ORDERS_1:5</a></i>;<br/>
<a NAME="E3:81_1_2"/><b>then </b>
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:81_1_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>)
 <b>by </b><i><a class="ref" href="modelc_1.html#D13" target="_self">Def13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:81_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:81_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>)
 
<b>by </b><i><a class="ref" href="modelc_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:81_1_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:81_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>) )
 <b>by </b><i><a class="ref" href="relset_2.html#T9">RELSET_2:9</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E12:81_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>) ) ) )
 <b>by </b><i>, </i>;<br/>


</div>
