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

<b>consider </b><font color="Maroon">B</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E2:1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S2">ALTCAT_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b</font>,<font color="Olive">c</font>) holds <br/><font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>,<font color="Olive">f</font>,<font color="Olive">g</font>) ) );<br/>
<a NAME="E3:1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T72">MCART_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font>, <font color="Maroon">c</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Olive">f</font>,<font color="Olive">g</font>) );<br/>
<a NAME="E5:1_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>)</span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>)</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>) &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>) &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">x2</font>,<font color="Maroon">x1</font>);<br/>

<b>thus </b><a NAME="E4:1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>)
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E5:1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:1_1_1"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>)</span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>) )
 <b>and </b><br/><a NAME="E8:1_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>)</span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E9:1_1_1"/>
( <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>) &amp; <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>) &amp; <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>) )
 
<b>by </b><i/>;<br/>
<a NAME="E10:1_1_1"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>)</span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>) )
 
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a>, <a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b><font color="Maroon">j</font> = <font color="Maroon">F</font>;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

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

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

<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E12:1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>

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

<b>assume </b><a NAME="E13:1_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>) &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>) )
 ;<br/>

<a NAME="E14:1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>)</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">f'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:1_1_1"/><i><font color="Green">E53</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">f'</font>,<font color="Maroon">g'</font>) )
 <b>by </b><i/>;<br/>
<a NAME="E17:1_1_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E18:1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">C</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>and </b><br/><a NAME="E6:1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font><b> such that </b><br/><a NAME="E3:1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>and </b><br/><a NAME="E4:1_1_2"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>) &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>) holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Olive">f</font>,<font color="Olive">g</font>)
 <b>by </b><i/>;<br/><a NAME="E5:1_1_2"/>
( <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:1_1_2"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>set </b><font color="Maroon">alt</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">B</font>,<font color="Maroon">C</font> #);<br/>
<a NAME="E10:1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">B</font>,<font color="Maroon">C</font> #) holds  <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E11:1_1"/>
 <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">B</font>,<font color="Maroon">C</font> #) is <a href="altcat_1.html#V2">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">B</font>,<font color="Maroon">C</font> #), <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">B</font>,<font color="Maroon">C</font> #), <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">B</font>,<font color="Maroon">C</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a><br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">o1</font>, <font color="Maroon">b</font> = <font color="Maroon">o2</font>, <font color="Maroon">c</font> = <font color="Maroon">o3</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>, <font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span>;<br/>
<b>assume </b><a NAME="E3:1_1_3"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E4:1_1_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
;<br/>
<a NAME="E5:1_1_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>) &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>) )
 
<b>by </b><i/>;<br/>
<a NAME="E6:1_1_3"/><b>then </b>
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:1_1_3"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">alt</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">B</font>,<font color="Maroon">C</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_1.html#L2">AltCatStr</a>  ;<br/>
<b>take </b>
<font color="Maroon">alt</font>
;<br/>

<b>thus </b><a NAME="E13:1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">alt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>thus </b><a NAME="E14:1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">alt</font> holds  <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>)
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">alt</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">alt</font>, <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">alt</font>;<br/>

<b>assume </b><a NAME="E15:1_1"/><i><font color="Green">E62</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E16:1_1"/>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">a'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">B</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font><b> such that </b><br/><a NAME="E18:1_1"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>and </b><br/><a NAME="E19:1_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a'</font>,<font color="Maroon">b'</font>) &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b'</font>,<font color="Maroon">c'</font>) holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Olive">f</font>,<font color="Olive">g</font>)
 <b>by </b><i/>;<br/>
<a NAME="E20:1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/>



<a NAME="E21:1_1"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>) )
 
<b>by </b><i/>;<br/>
<a NAME="E22:1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>)
 
<b>by </b><i>, , </i>;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">alt</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>)
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a>, </i>
;<br/>


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