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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>,<font color="Maroon">a<sub>5</sub></font>);<br/>
<a NAME="E1:4_1"/><i><font color="Green">E43</font></i>: 
for <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>()<br/> 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="Maroon">H<sub>1</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>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">c</font>)
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">C</font> being  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> <b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E4:4_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <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>and </b><br/><a NAME="E5:4_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>  st <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#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="Olive">b</font>,<font color="Olive">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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font><br/> for <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b</font>,<font color="Olive">c</font> holds  <font color="Olive">g</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</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>)
 <b>from </b><i><a class="ref" href="yellow18.html#S1" target="_self">YELLOW18:sch 1</a>();<br/></i>
<a NAME="E6:4_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Olive">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="Olive">b</font>,<font color="Olive">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Maroon">H<sub>1</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>),<font color="Olive">h</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">d</font>,<font color="Olive">f</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>,<font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">g</font>,<font color="Olive">h</font>))
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">d</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<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> , <font color="Maroon">h</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green">E54</font></i>: 
( <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">a</font>,<font color="Maroon">b</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">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">h</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">c</font>,<font color="Maroon">d</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 ;<br/>

<a NAME="E2:4_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>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</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">c</font>,<font color="Maroon">d</font>) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">H<sub>1</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>),<font color="Maroon">h</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">f</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font>))
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:4_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>  ex <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">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="Olive">a</font>,<font color="Olive">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( for <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>,<font color="Olive">g</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:4_1_2"/><i><font color="Green">E56</font></i>: 
<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">a</font>)
 <b>and </b><br/><a NAME="E3:4_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Olive">b</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">a</font>,<font color="Olive">b</font>,<font color="Maroon">f</font>,<font color="Olive">g</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E4:4_1_2"/>
<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">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, </i>;<br/>

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

<a NAME="E5:4_1_2"/>
<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>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:4_1_2"/>
for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>,<font color="Olive">g</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:4_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>  ex <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">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="Olive">a</font>,<font color="Olive">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( for <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="Olive">b</font>,<font color="Olive">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>,<font color="Olive">a</font>,<font color="Olive">a</font>,<font color="Olive">g</font>,<font color="Olive">f</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_3"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:4_1_3"/><i><font color="Green">E66</font></i>: 
<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">a</font>)
 <b>and </b><br/><a NAME="E3:4_1_3"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <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="Maroon">a</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font>,<font color="Olive">g</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E4:4_1_3"/>
<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">a</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, </i>;<br/>

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

<a NAME="E5:4_1_3"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</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">a</font>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:4_1_3"/>
for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font>,<font color="Olive">g</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:4_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">C</font> is <a href="altcat_1.html#V11">associative</a>
 
<b>from </b><i><a class="ref" href="yellow18.html#S2" target="_self">YELLOW18:sch 2</a>(, );<br/></i>
<a NAME="E10:4_1"/>
<font color="Maroon">C</font> is <a href="altcat_1.html#V12">with_units</a>
 
<b>from </b><i><a class="ref" href="yellow18.html#S3" target="_self">YELLOW18:sch 3</a>(, , );<br/></i>
<b>hence </b><a NAME="E11:4_1"/>
 ex <font color="Olive">C</font> being <a href="altcat_1.html#V6">strict</a> <a href="altcat_1.html#NM4">category</a> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">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>) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">C</font>  st <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#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="Olive">b</font>,<font color="Olive">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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font><br/> for <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b</font>,<font color="Olive">c</font> holds  <font color="Olive">g</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">f</font> <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>) ) )
 <b>by </b><i>, , , </i>;<br/>


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