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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:86_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E3:86_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E1:86"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:86_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds <font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E6:86_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E2:86"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D2" target="_self">YELLOW18:def 2</a><br/>

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E7:86_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/>( <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></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; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b<sub>1</sub></font>) holds <br/><font color="Olive">b<sub>2</sub></font> is <a href="altcat_3.html#V3">iso</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>)
<b>by </b><i><a class="txt" href="yellow18.html#E5:86_1"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>))
<b>by </b><i><a class="txt" href="yellow18.html#E2:86_1"><i><font color="Green">E47</font></i></a></i>
;<br/>
<a NAME="E2:86_1_1"/>
<span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="functor0.html#T30">FUNCTOR0:30</a></i>;<br/>
<a NAME="E3:86_1_1"/><b>then </b>
( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>) is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:86"><i><font color="Green">E43</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:86_1_1"/>
( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>) holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="altcat_3.html#V3">iso</a> ) )
 <b>by </b><i><a class="ref" href="yellow18.html#T43" target="_self">Th43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:86_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b<sub>1</sub></font>) holds <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b<sub>2</sub></font>) holds <br/><font color="Olive">b<sub>5</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:86_1_2"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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="E2:86_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E3:86_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>6</sub></font>) as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E5:86_1_2"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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><a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E6:86_1_2"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="functor3.html#T6">FUNCTOR3:6</a></i>;<br/>
<a NAME="E7:86_1_2"/>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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><a class="txt" href="yellow18.html#E2:86_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yellow18.html#E3:86_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<a NAME="E8:86_1_2"/><b>then </b>
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>)) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:86_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yellow18.html#E6:86_1_2"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>))
<b>by </b><i><a class="txt" href="yellow18.html#E3:86_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E5:86"><i><font color="Green">E45</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="functor0.html#T32">FUNCTOR0:32</a></i>
;<br/>
<b>hence </b><a NAME="E10:86_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>5</sub></font>) holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">c<sub>6</sub></font>) holds <br/><font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:86_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self">Th38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:86_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="functor3.html#M1">natural_equivalence</a> of <font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1">are_naturally_equivalent</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) )
 
<b>from </b><i><a class="ref" href="yellow18.html#S15" target="_self">YELLOW18:sch 15</a>(<a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="yellow18.html#E8:86_1"><i><font color="Green">E52</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:86_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1">are_naturally_equivalent</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E11:86_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/>( <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></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; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">b<sub>1</sub></font>) holds <br/><font color="Olive">b<sub>2</sub></font> is <a href="altcat_3.html#V3">iso</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>)
<b>by </b><i><a class="txt" href="yellow18.html#E2:86_1"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>))
<b>by </b><i><a class="txt" href="yellow18.html#E5:86_1"><i><font color="Green">E49</font></i></a></i>
;<br/>
<a NAME="E2:86_1_3"/>
<span class="p1">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="functor0.html#T30">FUNCTOR0:30</a></i>;<br/>
<a NAME="E3:86_1_3"/><b>then </b>
( <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E4:86"><i><font color="Green">E44</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_3"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:86_1_3"/>
( <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></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; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="altcat_3.html#V3">iso</a> ) )
 <b>by </b><i><a class="ref" href="yellow18.html#T43" target="_self">Th43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:86_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">b<sub>1</sub></font>) holds <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">b<sub>2</sub></font>) holds <br/><font color="Olive">b<sub>5</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<b>assume </b><a NAME="E1:86_1_4"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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="E2:86_1_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E4:86_1_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>8</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E5:86_1_4"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></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><a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>

<a NAME="E6:86_1_4"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="functor3.html#T6">FUNCTOR3:6</a></i>;<br/>
<a NAME="E7:86_1_4"/>
( <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></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><a class="txt" href="yellow18.html#E5:86_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yellow18.html#E6:86_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<a NAME="E8:86_1_4"/><b>then </b>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>)) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:86_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="yellow18.html#E6:86_1_4"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>)) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E2:86_1_4"><i><font color="Green">E56</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>8</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E6:86"><i><font color="Green">E46</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>8</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="functor0.html#T32">FUNCTOR0:32</a></i>
;<br/>
<b>hence </b><a NAME="E10:86_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>7</sub></font>) holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>8</sub></font>) holds <br/><font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow18.html#E4:86_1_4"><i><font color="Green">E57</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self">Th38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:86_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="functor3.html#M1">natural_equivalence</a> of  <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="functor3.html#R1">are_naturally_equivalent</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) )
 
<b>from </b><i><a class="ref" href="yellow18.html#S15" target="_self">YELLOW18:sch 15</a>(<a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow18.html#E12:86_1"><i><font color="Green">E54</font></i></a>);<br/></i>
<b>hence </b><a NAME="E14:86_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="functor3.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>, <a href="functor0.html#K12">id</a> <font color="Maroon">F<sub>2</sub></font>() <a href="functor3.html#R1">are_naturally_equivalent</a> 
 ;<br/>


</div>
