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

<a NAME="E1:23_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font><br/> for <font color="Olive">g1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K3">.</a> <font color="Olive">a</font></span>)</span>  st <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">a</font>) holds <br/>for <font color="Olive">g2</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Olive">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K3">.</a> <font color="Olive">b</font></span>)</span>  st <font color="Olive">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b</font>) holds <br/><font color="Olive">g2</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K6">.</a> <font color="Olive">f</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Olive">g1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:23_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">a</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"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Olive">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K3">.</a> <font color="Olive">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()<b> such that </b><br/><a NAME="E4:23_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>and </b><br/><a NAME="E5:23_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">a</font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S14" target="_self">YELLOW18:sch 14</a>(, );<br/></i>
<a NAME="E6:23_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a><br/>

<b>thus </b><a NAME="E1:23_1_1"/>
not <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:23_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() <a href="functor3.html#R1">are_naturally_equivalent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:23_1_2"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4">FUNCTOR3:def 4</a><br/>

<b>thus </b><a NAME="E2:23_1_2"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i/>;<br/>

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

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

<a NAME="E3:23_1_2"/>
<font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a</font>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:23_1_2"/>
<font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E1:23_1_3"/>
<font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a</font>)
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:23_1_3"/>
<font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E9:23_1"/><b>then </b>
<font color="Maroon">t</font> is    <a href="functor3.html#M1">natural_equivalence</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()
 
<b>by </b><i>, <a class="ref" href="functor3.html#D5">FUNCTOR3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:23_1"/>
 ex <font color="Olive">t</font> being   <a href="functor3.html#M1">natural_equivalence</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() st <br/>( <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() <a href="functor3.html#R1">are_naturally_equivalent</a>  &amp; ( for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Olive">t</font> <a href="functor2.html#K2">!</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">a</font>) ) )
 <b>by </b><i>, </i>;<br/>


</div>
