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<a NAME="E1:23_1"/><i><font color="Green" title="E8">A3</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"><font color="Maroon">F<sub>3</sub></font>() <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"><font color="Maroon">F<sub>4</sub></font>() <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>5</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"><font color="Maroon">F<sub>3</sub></font>() <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"><font color="Maroon">F<sub>4</sub></font>() <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>5</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"><font color="Maroon">F<sub>3</sub></font>() <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"><font color="Maroon">F<sub>4</sub></font>() <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>
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:23"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<a NAME="E2:23_1"/><i><font color="Green" title="E9">A4</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">F<sub>5</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"><font color="Maroon">F<sub>3</sub></font>() <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"><font color="Maroon">F<sub>4</sub></font>() <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>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:23"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c1">t</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">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" title="E10">A5</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor2.html#R2" title="FUNCTOR2:pred.2">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" title="E11">A6</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">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</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>)
 <b>from </b><i><a class="ref" href="yellow18.html#S14" target="_self" title="YELLOW18:sch.14">YELLOW18:sch 14</a>(<a class="txt" href="yellow18.html#E2:23_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="yellow18.html#E1:23_1"><i><font color="Green" title="E8">A3</font></i></a>);<br/></i>
<a NAME="E6:23_1"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="functor2.html#R1" title="FUNCTOR2:pred.1">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" title="c2">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>(); <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a><br/>

<b>thus </b><a NAME="E1:23_1_1"/>
not <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">F<sub>4</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow18.html#E1:23"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:23_1"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">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" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="yellow18.html#E4:23_1"><i><font color="Green" title="E10">A5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4" title="FUNCTOR3:def.4">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" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="txt" href="yellow18.html#E6:23_1"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c1">t</font>
;<br/>

<b>let </b><font color="Maroon" title="c2">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/>

<a NAME="E3:23_1_2"/>
<font color="Maroon" title="c1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c2">a</font>)
 
<b>by </b><i><a class="txt" href="yellow18.html#E5:23_1"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_2"/>
<font color="Maroon" title="c1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:23"><i><font color="Green" title="E6">A1</font></i></a></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" title="c2">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/><a NAME="E1:23_1_3"/>
<font color="Maroon" title="c1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c2">a</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E5:23_1"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E2:23_1_3"/>
<font color="Maroon" title="c1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:23"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:23_1"/><b>then </b>
<font color="Maroon" title="c1">t</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">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="txt" href="yellow18.html#E7:23_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="functor3.html#D5" title="FUNCTOR3:def.5">FUNCTOR3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:23_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">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" 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>5</sub></font>(<font color="Olive" title="b2">a</font>) ) )
 <b>by </b><i><a class="txt" href="yellow18.html#E5:23_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="yellow18.html#E7:23_1"><i><font color="Green" title="E13">A8</font></i></a></i>;<br/>


</div>
