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

<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E</font>,<font color="Maroon">E</font>;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="abian.html#NR4" target="_self">has_no_fixpoint</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>] <b>means </b>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K5">`1</a>  &amp; <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K6">`2</a>  &amp; <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K7">`3</a>  );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E</font>,<font color="Maroon">E</font> =  <a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:37"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">x0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:37_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x0</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> );<br/>
<b>set </b><font color="Maroon">c1</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span> ;<br/>
<a NAME="E4:37_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c1</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; not <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> );<br/>
<b>set </b><font color="Maroon">c2</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>] </span>}</span> ;<br/>
<a NAME="E6:37_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c2</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c3</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
( <font color="Maroon">c1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c2</font> or <font color="Maroon">c1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c2</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">b</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</font></span><a href="domain_1.html#K4">]</a></span>;<br/><a NAME="E2:37_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c3</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>;<br/><b>thus </b><a NAME="E3:37_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:37_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c3</font> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp;  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> ) )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp;  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; not <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> ) )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c3</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">xc</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> ;<br/>
<a NAME="E4:37_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">xc</font>,<font color="Maroon">x0</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:37_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:37_1_1_1_1"/><i><font color="Green">E61</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_2"/>
( <font color="Maroon">k</font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">k</font> is <a href="abian.html#V1" target="_self">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>thus </b><a NAME="E3:37_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">xc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:37_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_1_1_1_2"/>
not <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xc</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/><a NAME="E4:37_1_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">xc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 ;<br/><b>hence </b><a NAME="E5:37_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_2"/><i><font color="Green">E64</font></i>: 
not <font color="Maroon">k</font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>thus </b><a NAME="E3:37_1_1_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_2_2_1_1"/>
( <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">xc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:37_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_2_1_1_2"/>
not <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xc</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/><a NAME="E4:37_1_1_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">xc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 ;<br/><b>hence </b><a NAME="E5:37_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div><b>thus </b><a NAME="E4:37_1_1_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_1_2"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:37_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_1_2_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E5:37_1_1_1_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp;  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">yc</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:37_1_1_1_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font>, <font color="Maroon">l</font> = <font color="Maroon">l</font> as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">l2</font> = <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E13:37_1_1_1_2_1"/><i><font color="Green">E71</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E72</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</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_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font></span>)</span>
<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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></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_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/>
<a NAME="E16:37_1_1_1_2_1"/><b>then </b>
<font color="Maroon">yc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E17:37_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_1_2"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:37_1_1_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_1_3"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:37_1_1_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_1_3_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E5:37_1_1_1_3_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp;  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; not <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">yc</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:37_1_1_1_3_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; not <font color="Maroon">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">l</font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">l2</font> = <font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E14:37_1_1_1_3_1"/><i><font color="Green">E92</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E93</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</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_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font></span>)</span>
<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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></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_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/>
<a NAME="E17:37_1_1_1_3_1"/><b>then </b>
<font color="Maroon">yc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E18:37_1_1_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_1_3"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:37_1_1_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2"/>
<font color="Maroon">c1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">x11</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E5:37_1_1_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x11</font> &amp;  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x12</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E7:37_1_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x12</font> &amp;  ex <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; not <font color="Olive">k</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">l</font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:37_1_1_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">k1</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">l1</font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:37_1_1_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> &amp; not <font color="Maroon">k2</font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">l2</font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i/>;<br/><a NAME="E13:37_1_1_2"/><i><font color="Green">E103</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green">E104</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
;<br/><i><font color="Green">E105</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l2</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">l1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
;<br/><a NAME="E16:37_1_1_2"/>
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">k</font> being <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font>, <font color="Maroon">l1</font> = <font color="Maroon">l1</font>, <font color="Maroon">l2</font> = <font color="Maroon">l2</font> as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E3:37_1_1_2_3"/><i><font color="Green">E107</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_2_3_1"/>
( <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font> or <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_2_3_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>;<br/><b>reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>take </b>
<font color="Maroon">k</font>
;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font></span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E5:37_1_1_2_3_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><b>thus </b><a NAME="E6:37_1_1_2_3_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2_3_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>;<br/><b>reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>take </b>
<font color="Maroon">k</font>
;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font></span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E5:37_1_1_2_3_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><b>thus </b><a NAME="E6:37_1_1_2_3_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">k</font> being  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:37_1_1_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> )
 ;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i/>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> ;<br/><a NAME="E20:37_1_1_2"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>;<br/>
<a NAME="E1:37_1_1_2_4"/>
<span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">l</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>hence </b><a NAME="E2:37_1_1_2_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">Odl</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c</font>, <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:37_1_1_2"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> holds  <font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_2_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Odl</font>
 ;<br/><b>then reconsider </b><font color="Maroon">nc</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:37_1_1_2_5"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">nc</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">nc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_1_1_2_5"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E6:37_1_1_2_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">nc</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T30">EQREL_1:30</a></i>;<br/><b>then consider </b><font color="Maroon">kn</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ln</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:37_1_1_2_5"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">kn</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">nc</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">ln</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>;<br/><a NAME="E9:37_1_1_2_5"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i/>;<br/><div><i><font color="Green">E123</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_2_5_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">i</font>]
 ;<br/><a NAME="E2:37_1_1_2_5_1"/><i><font color="Green">E125</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> 1</span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:37_1_1_2_5_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><a NAME="E11:37_1_1_2_5"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E12:37_1_1_2_5"/>
2 <a href="nat_1.html#K2">*</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E13:37_1_1_2_5"/><b>then </b><i><font color="Green">E127</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><b>set </b><font color="Maroon">dk</font> = <font color="Maroon">ln</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">mk</font> = <font color="Maroon">ln</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">k</font>;<br/><a NAME="E14:37_1_1_2_5"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">ln</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">ln</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">ln</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E15:37_1_1_2_5"/>
<font color="Maroon">ln</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">kmk</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">ln</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">k</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E17:37_1_1_2_5"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">ln</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">kmk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ln</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E18:37_1_1_2_5"/><i><font color="Green">E133</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">kn</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ln</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">kmk</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">kn</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">nc</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">kmk</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">kn</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">nc</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">kmk</font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ln</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, <a class="ref" href="abian.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">kmk</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ln</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="abian.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ln</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i/>
;<br/><a NAME="E20:37_1_1_2_5"/><b>then </b>
<font color="Maroon">kn</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">kmk</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Odl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E21:37_1_1_2_5"/>
not <font color="Maroon">Odl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Odl</font> = <font color="Maroon">Odl</font> as  <a href="relat_1.html#V2">non-empty</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c</font>, <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">odl</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c</font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E26:37_1_1_2"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> holds  <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">odl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a>;<br/><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>not <font color="Maroon">odl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a>;<br/><b>set </b><font color="Maroon">c1</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span> ;<br/><b>set </b><font color="Maroon">d2</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>] </span>}</span> ;<br/><b>set </b><font color="Maroon">d1</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E27:37_1_1_2"/><i><font color="Green">E139</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E28:37_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">d1</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E30:37_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i><b>then reconsider </b><font color="Maroon">d2</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">d3</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>take </b><font color="Maroon">b</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font>,<font color="Maroon">d3</font></span><a href="domain_1.html#K4">]</a></span>;<br/><a NAME="E33:37_1_1_2"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> &amp; <font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d3</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>;<br/><a NAME="E34:37_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(0) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E35:37_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/><a NAME="E36:37_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E37:37_1_1_2"/><b>then </b>
 <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E38:37_1_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i/>;<br/><a NAME="E39:37_1_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 ;<br/><i><font color="Green">E144</font></i>: <font color="Maroon">d1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">d3</font> = 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">d3</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><a NAME="E41:37_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_7"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:37_1_1_2_7_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">d2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_2_7_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_2_7_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">xc</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E3:37_1_1_2_7_1_1_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">xc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> is <a href="abian.html#V1" target="_self">even</a> )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_2_7_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2_7_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">xc</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E3:37_1_1_2_7_1_1_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">xc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> is <a href="abian.html#V1" target="_self">even</a> )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_2_7_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37_1_1_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">xc</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> ;<br/>
<a NAME="E4:37_1_1_2_7"/>
( <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> is <a href="abian.html#V1" target="_self">even</a> )
 
;<br/>
<a NAME="E5:37_1_1_2_7"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d2</font> )
 
;<br/>
<b>hence </b><a NAME="E6:37_1_1_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">d2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E42:37_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E43:37_1_1_2"/><i><font color="Green">E148</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">d2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_2_8"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">d2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_8"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:37_1_1_2_8"/>
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> is <a href="abian.html#V1" target="_self">even</a> ) &amp;  ex <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; not <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> is <a href="abian.html#V1" target="_self">even</a> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:37_1_1_2_8"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E44:37_1_1_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">d1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">d2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E45:37_1_1_2"/><i><font color="Green">E151</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">odl</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font>;<br/>

<b>assume </b><a NAME="E1:37_1_1_2_9"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/>

<div><i><font color="Green">E153</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_2_9_1"/>
<font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:37_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:37_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/><a NAME="E4:37_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E5:37_1_1_2_9_1"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 ;<br/><b>hence </b><a NAME="E6:37_1_1_2_9_1"/>
contradiction
 <b>by </b><i><a class="ref" href="abian.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ofx</font> = <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font>, <font color="Maroon">ox</font> = <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E5:37_1_1_2_9"/>
<font color="Maroon">ox</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ox1</font> = <font color="Maroon">ox</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:37_1_1_2_9"/><i><font color="Green">E158</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E159</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ox1</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ox1</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ox1</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ox</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>

;<br/>
<a NAME="E9:37_1_1_2_9"/>
<font color="Maroon">ox</font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ox</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E11:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ox</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E12:37_1_1_2_9"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ox</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E13:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ox1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="abian.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E14:37_1_1_2_9"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">ox1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:37_1_1_2_9"/>
<font color="Maroon">ofx</font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:37_1_1_2_9"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">ofx</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ox</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E2:37"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<i><font color="Green">E162</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ofx</font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ofx</font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">ofx</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font>
<b>by </b><i><a class="txt" href="abian.html#E1:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E18:37_1_1_2_9"/>
<font color="Maroon">ofx</font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E19:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ofx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E20:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ofx</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E21:37_1_1_2_9"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ofx</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E22:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ofx</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E23:37_1_1_2_9"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">ofx</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E24:37_1_1_2_9"/>
<font color="Maroon">ox</font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">Odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E25:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ofx</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ox</font>
 
<b>by </b><i>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E26:37_1_1_2_9"/><b>then </b>
<font color="Maroon">ofx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ox</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E27:37_1_1_2_9"/>
<font color="Maroon">odl</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="abian.html#E2:37_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E46:37_1_1_2"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_10"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_2_10"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:37_1_1_2_10_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_10_1"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:37_1_1_2_10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="abian.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E6:37_1_1_2_10_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp; <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> is <a href="abian.html#V1" target="_self">even</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">ox</font> = <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> as  <a href="abian.html#V1" target="_self">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">yc</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i><a class="ref" href="abian.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E9:37_1_1_2_10_1"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xx</font>
 
<b>by </b><i><a class="ref" href="abian.html#D1" target="_self">Def1</a>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E10:37_1_1_2_10_1"/><b>then </b>
<font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">yc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ox</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="abian.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E11:37_1_1_2_10_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_2_10"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E47:37_1_1_2"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_11"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_2_11"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d2</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:37_1_1_2_11_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_11_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E5:37_1_1_2_11_1"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp; not <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i><a class="ref" href="abian.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ox</font> = <font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> as  <a href="abian.html#V1" target="_self">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">yc</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> <b>by </b><i><a class="ref" href="abian.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E8:37_1_1_2_11_1"/>
<font color="Maroon">odl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">yc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ox</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, , <a class="ref" href="abian.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E9:37_1_1_2_11_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_2_11"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E48:37_1_1_2"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_2_12"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_12"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="mcart_1.html#K7">`3</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:37_1_1_2_12"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="abian.html#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37_1_1_2_12"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="abian.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:37_1_1_2_12"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="abian.html#E1:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:37_1_1_2_12"/><b>then </b>
<font color="Maroon">r</font> <a href="abian.html#R2" target="_self">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="abian.html#E1:37"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:37_1_1_2_12"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">E</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green">E171</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">E1c</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">E1</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">E2c</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">E2</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">E3c</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">E3</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> ;<br/>
<b>take </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
;<br/>

<b>take </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
;<br/>

<b>take </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
;<br/>

<b>thus </b><a NAME="E5:37"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:37_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 ;<br/><a NAME="E2:37_2_1"/><b>then </b><i><font color="Green">E175</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_1_1"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E5:37_2_1_1_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:37_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_1_2"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E5:37_2_1_1_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:37_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_3"><b>suppose </b></a><a NAME="E1:37_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_1_3"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E5:37_2_1_1_3"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:37_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/>

<b>set </b><font color="Maroon">c</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>;<br/>
<a NAME="E3:37_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E5:37_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:37_2"/><b>then </b><i><font color="Green">E185</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2_1"><b>suppose </b></a><a NAME="E1:37_2_2_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 ;<br/><div class="add"><a NAME="E2:37_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:37_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:37_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:37_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2_2"><b>suppose </b></a><a NAME="E1:37_2_2_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 ;<br/><div class="add"><a NAME="E2:37_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:37_2_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:37_2_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:37_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2_3"><b>suppose </b></a><a NAME="E1:37_2_2_3"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 ;<br/><div class="add"><a NAME="E2:37_2_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:37_2_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:37_2_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E6:37"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_3"/>
not <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_3"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E7:37_3"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_3"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">YY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:37_3"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font> &amp; <font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">cc</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E13:37_3"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">YY</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i/>;<br/>
<a NAME="E14:37_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:37_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:37_3"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:37_3"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E19:37_3"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E20:37_3"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:37_3"/><b>then </b>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:37_3"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cc</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">xx'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:37_3"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">xx'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">cc</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E25:37_3"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">cc</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E26:37_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:37_3"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> = 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E29:37_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">xx</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E30:37_3"/><b>then </b><i><font color="Green">E205</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>
<a NAME="E31:37_3"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">YY</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E32:37_3"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">YY</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">YY</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E33:37_3"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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

<b>thus </b><a NAME="E7:37"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_4"/>
not <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_4"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_4"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E7:37_4"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_4"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">YY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:37_4"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font> &amp; <font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">cc</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E13:37_4"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">YY</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i/>;<br/>
<a NAME="E14:37_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:37_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:37_4"/><b>then </b><i><font color="Green">E213</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:37_4"/><i><font color="Green">E214</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E19:37_4"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E20:37_4"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:37_4"/><b>then </b>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:37_4"/><b>then </b><i><font color="Green">E216</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cc</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">xx'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:37_4"/><i><font color="Green">E217</font></i>: 
( <font color="Maroon">xx'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">cc</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E25:37_4"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">cc</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E26:37_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:37_4"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> = 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E29:37_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">xx</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E30:37_4"/><b>then </b><i><font color="Green">E220</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>
<a NAME="E31:37_4"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">YY</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E32:37_4"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">YY</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">YY</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E33:37_4"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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

<b>thus </b><a NAME="E8:37"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_5"/>
not <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_5"/><i><font color="Green">E222</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_5"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E7:37_5"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_5"/><i><font color="Green">E225</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">YY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:37_5"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font> &amp; <font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">cc</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E13:37_5"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">YY</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i/>;<br/>
<a NAME="E14:37_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:37_5"/><b>then </b><i><font color="Green">E228</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:37_5"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E18:37_5"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E19:37_5"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">cc</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:37_5"/><b>then </b><i><font color="Green">E231</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cc</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">xx'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:37_5"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">xx'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">cc</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx'</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E23:37_5"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">cc</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E24:37_5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:37_5"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> = 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E27:37_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">xx</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E28:37_5"/><b>then </b><i><font color="Green">E235</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>
<a NAME="E29:37_5"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">YY</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E30:37_5"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">YY</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">YY</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E31:37_5"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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


</div>
