<?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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b</font>,<font color="Olive">c</font>] &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Olive">c</font></span><a href="domain_1.html#K1">]</a></span> );<br/>
<a NAME="E1:10_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:10_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>]
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">u</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>;<br/>

<b>thus </b><a NAME="E4:10_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>thus </b><a NAME="E5:10_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">u</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E4:10_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E5:10_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<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>  =  <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>;<br/>
<a NAME="E6:10_1"/><i><font color="Green">E46</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> 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="10_1_2"><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="E1:10_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:10_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E3:10_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:10_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_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="E1:10_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_2_1_1_1"/>
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_2_1_1_2"/>
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1_2"/>
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>,0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E8:10_1"/><i><font color="Green">E68</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> 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>
<a NAME="E9:10_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">e</font>,0,<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">e</font>,1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:10_1_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>]
 <b>and </b><br/><a NAME="E4:10_1_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E5:10_1_3"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:10_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><i><font color="Green">E74</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,0</span><a href="tarski.html#K4">]</a></span> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<a NAME="E8:10_1_3"/><i><font color="Green">E75</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E9:10_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,1</span><a href="tarski.html#K4">]</a></span> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E11:10_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">e</font>,0,<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">e</font>,1]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 2;<br/>
<b>consider </b><font color="Maroon">D</font> being   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E11:10_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>and </b><br/><a NAME="E12:10_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">d</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> holds <br/> (  <a href="trees_2.html#K1">succ</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Olive">d</font>) </span>}</span>  &amp; ( 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="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Olive">d</font>) holds <br/><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Olive">d</font></span>)</span>,<font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="trees_2.html#S9">TREES_2:sch 9</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:10_1_4"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font></span>)</span>
 ;<br/><a NAME="E2:10_1_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</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="E1:10_1_4_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:10_1_4_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:10_1_4_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/>
<a NAME="E5:10_1_4_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T71">NAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_4_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">v</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="E2:10_1_4_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E3:10_1_4_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_4_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span> 
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:10_1_4"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E14:10_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> is <a href="bintree1.html#V1">binary</a>
 
<b>by </b><i><a class="ref" href="bintree1.html#D2">BINTREE1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  <a href="bintree1.html#V2">binary</a> <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="bintree1.html#D3">BINTREE1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font>;<br/><a NAME="E1:10_1_5"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1_5_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</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="E1:10_1_5_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:10_1_5_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:10_1_5_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/>
<a NAME="E5:10_1_5_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T71">NAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_5_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">v</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="E2:10_1_5_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E3:10_1_5_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_5_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 </span>}</span> 
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:10_1_5"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:10_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E18:10_1"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font>;<br/>

<a NAME="E19:10_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,0,<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,1]
 
<b>by </b><i/>;<br/>
<a NAME="E20:10_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,1]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E21:10_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>]
 <b>by </b><i/>;<br/>


</div>
