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

<b>consider </b><font color="Maroon" title="c1">F</font> being  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">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" title="E47">A7</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>and </b><br/><a NAME="E3:86_1"/><i><font color="Green" title="E48">A8</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Maroon" title="c1">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E1:86"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">G</font> being  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">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" title="E49">A9</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>and </b><br/><a NAME="E6:86_1"/><i><font color="Green" title="E50">A10</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Maroon" title="c2">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E2:86"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c1">F</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D2" target="_self" title="YELLOW18:def.2">YELLOW18:def 2</a><br/>

<b>take </b>
<font color="Maroon" title="c2">G</font>
;<br/>

<b>set </b><font color="Maroon" title="c3">GF</font> = <font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>;<br/>
<b>set </b><font color="Maroon" title="c4">I</font> =  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E7:86_1"/><i><font color="Green" title="E51">A11</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b2">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/><font color="Olive" title="b2">f</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">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" title="c5">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<i><font color="Green" title="E52">A12</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
<font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T34" title="FUNCTOR0:th.34">FUNCTOR0:34</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>)
<b>by </b><i><a class="txt" href="yellow18.html#E5:86_1"><i><font color="Green" title="E49">A9</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">a</font>))
<b>by </b><i><a class="txt" href="yellow18.html#E2:86_1"><i><font color="Green" title="E47">A7</font></i></a></i>
;<br/>
<a NAME="E2:86_1_1"/>
<span class="p1">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="functor0.html#T30" title="FUNCTOR0:th.30">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" title="c5">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:86"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_1"><i><font color="Green" title="E52">A12</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" title="c5">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/><font color="Olive" title="b1">f</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )
 <b>by </b><i><a class="ref" href="yellow18.html#T43" target="_self" title="YELLOW18:th.43">Th43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:86_1"/><i><font color="Green" title="E52">A13</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b4">g1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/>for <font color="Olive" title="b5">g2</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>  st <font color="Olive" title="b5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">b</font>) holds <br/><font color="Olive" title="b5">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">g1</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" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">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" title="E53">A14</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:86_1_2"/><i><font color="Green" title="E54">A15</font></i>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>
<a NAME="E3:86_1_2"/><i><font color="Green" title="E55">A16</font></i>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">g2</font> = <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <b>by </b><i><a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>
<a NAME="E5:86_1_2"/><i><font color="Green" title="E56">A17</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">FUNCTOR0:def 19</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>;<br/>

<a NAME="E6:86_1_2"/><i><font color="Green" title="E57">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="functor3.html#T6" title="FUNCTOR3:th.6">FUNCTOR3:6</a></i>;<br/>
<a NAME="E7:86_1_2"/>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:86_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="yellow18.html#E3:86_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">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" title="c5">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:86_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="yellow18.html#E6:86_1_2"><i><font color="Green" title="E57">A18</font></i></a></i>;<br/>
<b>then </b><font color="Maroon" title="c7">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font></span>)</span> = 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">f</font>))
<b>by </b><i><a class="txt" href="yellow18.html#E3:86_1_2"><i><font color="Green" title="E55">A16</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_2"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self" title="YELLOW18:th.38">Th38</a></i>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E5:86"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green" title="E53">A14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="functor0.html#T32" title="FUNCTOR0:th.32">FUNCTOR0:32</a></i>
;<br/>
<b>hence </b><a NAME="E10:86_1_2"/>
for <font color="Olive" title="b1">g1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/>for <font color="Olive" title="b2">g2</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>  st <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) holds <br/><font color="Olive" title="b2">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">g1</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:86_1_2"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_2"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self" title="YELLOW18:th.38">Th38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:86_1"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; ( for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Olive" title="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">a</font>) ) )
 
<b>from </b><i><a class="ref" href="yellow18.html#S15" target="_self" title="YELLOW18:sch.15">YELLOW18:sch 15</a>(<a class="txt" href="yellow18.html#E7:86_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="txt" href="yellow18.html#E8:86_1"><i><font color="Green" title="E52">A13</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:86_1"/>
<font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c5">I</font> =  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c6">FG</font> = <font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font>;<br/>
<a NAME="E11:86_1"/><i><font color="Green" title="E53">A19</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b2">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/><font color="Olive" title="b2">f</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">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" title="c7">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<i><font color="Green" title="E54">A20</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> = 
<font color="Maroon" title="c1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T34" title="FUNCTOR0:th.34">FUNCTOR0:34</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>)
<b>by </b><i><a class="txt" href="yellow18.html#E2:86_1"><i><font color="Green" title="E47">A7</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>))
<b>by </b><i><a class="txt" href="yellow18.html#E5:86_1"><i><font color="Green" title="E49">A9</font></i></a></i>
;<br/>
<a NAME="E2:86_1_3"/>
<span class="p1">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="ref" href="functor0.html#T30" title="FUNCTOR0:th.30">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" title="c7">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E4:86"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_3"><i><font color="Green" title="E54">A20</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" title="c7">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/><font color="Olive" title="b1">f</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )
 <b>by </b><i><a class="ref" href="yellow18.html#T43" target="_self" title="YELLOW18:th.43">Th43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:86_1"/><i><font color="Green" title="E54">A21</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b4">g1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/>for <font color="Olive" title="b5">g2</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>  st <font color="Olive" title="b5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b2">b</font>) holds <br/><font color="Olive" title="b5">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">g1</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" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">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" title="E55">A22</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:86_1_4"/><i><font color="Green" title="E56">A23</font></i>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g1</font> = <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <b>by </b><i><a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/>
<a NAME="E4:86_1_4"/><i><font color="Green" title="E57">A24</font></i>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/>
<a NAME="E5:86_1_4"/><i><font color="Green" title="E58">A25</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">FUNCTOR0:def 19</a></i>;<br/>
<b>let </b><font color="Maroon" title="c10">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>;<br/>

<a NAME="E6:86_1_4"/><i><font color="Green" title="E59">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="functor3.html#T6" title="FUNCTOR3:th.6">FUNCTOR3:6</a></i>;<br/>
<a NAME="E7:86_1_4"/>
( <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E5:86_1"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="yellow18.html#E6:86_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">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" title="c7">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:86_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="yellow18.html#E6:86_1_4"><i><font color="Green" title="E59">A26</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c9">g1</font> = 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">f</font>)) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E2:86_1_4"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_4"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self" title="YELLOW18:th.38">Th38</a></i>
<br/>.= 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E6:86"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green" title="E55">A22</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:86_1_4"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="functor0.html#T32" title="FUNCTOR0:th.32">FUNCTOR0:32</a></i>
;<br/>
<b>hence </b><a NAME="E10:86_1_4"/>
for <font color="Olive" title="b1">g1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/>for <font color="Olive" title="b2">g2</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>  st <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) holds <br/><font color="Olive" title="b2">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">g1</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E4:86_1_4"><i><font color="Green" title="E57">A24</font></i></a>, <a class="txt" href="yellow18.html#E5:86_1_4"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self" title="YELLOW18:th.38">Th38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:86_1"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font> st <br/>(  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; ( for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Olive" title="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b2">a</font>) ) )
 
<b>from </b><i><a class="ref" href="yellow18.html#S15" target="_self" title="YELLOW18:sch.15">YELLOW18:sch 15</a>(<a class="txt" href="yellow18.html#E11:86_1"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="yellow18.html#E12:86_1"><i><font color="Green" title="E54">A21</font></i></a>);<br/></i>
<b>hence </b><a NAME="E14:86_1"/>
<font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font>, <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon">F<sub>2</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 ;<br/>


</div>
