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

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>

<b>set </b><font color="Maroon" title="c2">F</font> =  <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E1:15"/>
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">c'</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>  st  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b2">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b1">c</font></span>)</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">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b2">c'</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b1">c</font> holds <br/> (  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c'</font> <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;  ex <font color="Olive" title="b4">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c'</font> st <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">c'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E11">A1</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font></span>)</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/>

<b>let </b><font color="Maroon" title="c5">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font>;<br/>

<a NAME="E2:15_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:15_1"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>, <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c8">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font><b> such that </b><br/><a NAME="E5:15_1"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F2</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></i>;<br/>
<a NAME="E6:15_1"/><i><font color="Green" title="E13">A3</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/>
<a NAME="E7:15_1"/><i><font color="Green" title="E14">A4</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E8:15_1"/><i><font color="Green" title="E15">A5</font></i>: 
( <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font> &amp; <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E9:15_1"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">d</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E11:15_1"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c10">X</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>;<br/>
<i><font color="Green" title="E16">A6</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> = 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E2:15_1_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_1_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c11">y</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:15_1_1_2"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/>
<a NAME="E4:15_1_1_2"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/>
<a NAME="E5:15_1_1_2"/><i><font color="Green" title="E18">A8</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c12">z</font> <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="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">zz</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c12">z</font> ;<br/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">z</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c12">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c12">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c12">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c12">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c12">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c12">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">z</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c13">zz</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_1_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">z</font></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T15" title="NATTRA_1:th.15">NATTRA_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T21" title="NATTRA_1:th.21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E8:15_1_1_2"/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:15_1_1_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15_1_1"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E3:15_1_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">d</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E5:15_1_1"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">d</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:15_1"/><b>then </b><i><font color="Green" title="E16">A10</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T31" title="OPPCAT_1:th.31">OPPCAT_1:31</a></i>;<br/>
<a NAME="E13:15_1"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">d1</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E15:15_1"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c10">d1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c11">X</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>;<br/>
<i><font color="Green" title="E17">A11</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> = 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E2:15_1_2"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_2_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">z</font> = <font color="Maroon" title="c12">y</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:15_1_2_2"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/>
<a NAME="E4:15_1_2_2"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/>
<a NAME="E5:15_1_2_2"/><i><font color="Green" title="E19">A13</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> <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="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">zz</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> ;<br/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">z</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c13">z</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">zz</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_2_2"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">z</font></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T15" title="NATTRA_1:th.15">NATTRA_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#T21" title="NATTRA_1:th.21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E8:15_1_2_2"/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:15_1_2_2"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15_1_2"/><b>then </b><i><font color="Green" title="E18">A14</font></i>: 
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E3:15_1_2"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c10">d1</font>
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E5:15_1_2"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c10">d1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:15_1"/><b>then </b><i><font color="Green" title="E17">A15</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d1</font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T31" title="OPPCAT_1:th.31">OPPCAT_1:31</a></i>;<br/>
<i><font color="Green" title="E18">A16</font></i>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c'</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></i>
;<br/>
<i><font color="Green" title="E19">A17</font></i>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c'</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c'</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c4">c'</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></i>
;<br/>
<a NAME="E19:15_1"/><i><font color="Green" title="E20">A18</font></i>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d1</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E16:15_1"><i><font color="Green" title="E17">A15</font></i></a></i>;<br/>
<a NAME="E20:15_1"/><i><font color="Green" title="E21">A19</font></i>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E12:15_1"><i><font color="Green" title="E16">A10</font></i></a></i>;<br/>
<a NAME="E21:15_1"/><b>then </b><i><font color="Green" title="E22">A20</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c8">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <b>by </b><i><a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yoneda_1.html#E16:15_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green" title="E21">A19</font></i></a></i>;<br/>
<a NAME="E23:15_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</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="yoneda_1.html#E21:15_1"><i><font color="Green" title="E22">A20</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E24:15_1"/><b>then </b><i><font color="Green" title="E23">A21</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<a NAME="E25:15_1"/><b>then </b><i><font color="Green" title="E24">A22</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<a NAME="E26:15_1"/><i><font color="Green" title="E25">A23</font></i>: 
 <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ens_1.html#D14" title="ENS_1:def.14">ENS_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">t1</font> = <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<a NAME="E28:15_1"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> <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="yoneda_1.html#E18:15_1"><i><font color="Green" title="E19">A17</font></i></a>, <a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/>
<a NAME="E29:15_1"/><b>then </b>
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</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="yoneda_1.html#E25:15_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>;<br/>
<a NAME="E30:15_1"/><b>then </b><i><font color="Green" title="E26">A24</font></i>: 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font> <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="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>;<br/>
<a NAME="E31:15_1"/>
<font color="Maroon" title="c12">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T19" title="ENS_1:th.19">ENS_1:19</a></i>;<br/>
<a NAME="E32:15_1"/><b>then </b><i><font color="Green" title="E27">A25</font></i>: 
<font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T20" title="ENS_1:th.20">ENS_1:20</a></i>;<br/>
<a NAME="E33:15_1"/><b>then </b><i><font color="Green" title="E28">A26</font></i>: 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font> <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="yoneda_1.html#E30:15_1"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="funct_2.html#T14" title="FUNCT_2:th.14">FUNCT_2:14</a></i>;<br/>
<a NAME="E34:15_1"/><b>then </b><i><font color="Green" title="E29">A27</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <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="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>;<br/>
<a NAME="E35:15_1"/><i><font color="Green" title="E30">A28</font></i>: 
the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">t1</font> <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="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="yoneda_1.html#E33:15_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>;<br/>
<a NAME="E36:15_1"/><i><font color="Green" title="E31">A29</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</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="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="yoneda_1.html#E34:15_1"><i><font color="Green" title="E29">A27</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>;<br/>
<a NAME="E37:15_1"/><i><font color="Green" title="E32">A30</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> <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="yoneda_1.html#E25:15_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="yoneda_1.html#E35:15_1"><i><font color="Green" title="E30">A28</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E38:15_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c'</font> <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="yoneda_1.html#E17:15_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="yoneda_1.html#E24:15_1"><i><font color="Green" title="E23">A21</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1"><i><font color="Green" title="E31">A29</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>

<i><font color="Green" title="E33">A31</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
;<br/>
<a NAME="E41:15_1"/><b>then </b>
<font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font>,<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="yoneda_1.html#E32:15_1"><i><font color="Green" title="E27">A25</font></i></a>, <a class="txt" href="yoneda_1.html#E39:15_1"><i><font color="Green" title="E33">A31</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E42:15_1"/><b>then </b><i><font color="Green" title="E34">A32</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c4">c'</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c'</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="yoneda_1.html#E18:15_1"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1"><i><font color="Green" title="E31">A29</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E43:15_1"/><b>then </b>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/>
<a NAME="E44:15_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E45:15_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c'</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E42:15_1"><i><font color="Green" title="E34">A32</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c'</font> <b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">f</font>
;<br/>

<a NAME="E47:15_1"/><i><font color="Green" title="E35">A33</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c'</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="yoneda_1.html#E37:15_1"><i><font color="Green" title="E32">A30</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E48:15_1"/><i><font color="Green" title="E36">A34</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green" title="E21">A19</font></i></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>
<a NAME="E49:15_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c13">f</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E1:15_1_7"/><i><font color="Green" title="E37">A35</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</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="yoneda_1.html#E48:15_1"><i><font color="Green" title="E36">A34</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E2:15_1_7"/><b>then </b><i><font color="Green" title="E38">A36</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">ta1</font> = <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>
<i><font color="Green" title="E39">A37</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
;<br/>
<i><font color="Green" title="E40">A38</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
;<br/>
<i><font color="Green" title="E41">A39</font></i>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c'</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c'</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></i>
;<br/>
<i><font color="Green" title="E42">A40</font></i>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></i>
;<br/>
<a NAME="E8:15_1_7"/><i><font color="Green" title="E43">A41</font></i>: 
<font color="Maroon" title="c15">ta1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_7"><i><font color="Green" title="E38">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7"><i><font color="Green" title="E39">A37</font></i></a>, <a class="txt" href="yoneda_1.html#E5:15_1_7"><i><font color="Green" title="E40">A38</font></i></a>, <a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></i>;<br/>
<a NAME="E9:15_1_7"/>
<font color="Maroon" title="c15">ta1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T19" title="ENS_1:th.19">ENS_1:19</a></i>;<br/>
<a NAME="E10:15_1_7"/><b>then </b><i><font color="Green" title="E44">A42</font></i>: 
<font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T20" title="ENS_1:th.20">ENS_1:20</a></i>;<br/>
<a NAME="E11:15_1_7"/><b>then </b><i><font color="Green" title="E45">A43</font></i>: 
<font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font>, <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E12:15_1_7"/><i><font color="Green" title="E46">A44</font></i>: 
<font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>, <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:15_1_7"><i><font color="Green" title="E39">A37</font></i></a>, <a class="txt" href="yoneda_1.html#E5:15_1_7"><i><font color="Green" title="E40">A38</font></i></a>, <a class="txt" href="yoneda_1.html#E10:15_1_7"><i><font color="Green" title="E44">A42</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E13:15_1_7"/><i><font color="Green" title="E47">A45</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_7_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_7_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_7_5_1"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_7_5_1_1"><b>suppose </b></a><a NAME="E1:15_1_7_5_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_7_5_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_7"><i><font color="Green" title="E38">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7"><i><font color="Green" title="E39">A37</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_7"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="yoneda_1.html#E11:15_1_7"><i><font color="Green" title="E45">A43</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_7_5_1_2"><b>suppose </b></a><a NAME="E1:15_1_7_5_1_2"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font> <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/><div class="add"><a NAME="E2:15_1_7_5_1_2"/><b>then </b>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <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="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></i>;<br/><b>hence </b><a NAME="E3:15_1_7_5_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="yoneda_1.html#E2:15_1_7"><i><font color="Green" title="E38">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_7"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1_7"><i><font color="Green" title="E42">A40</font></i></a>, <a class="txt" href="yoneda_1.html#E12:15_1_7"><i><font color="Green" title="E46">A44</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:15_1_7"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_7_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_7_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_7_6_1"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_7_6_1_1"><b>suppose </b></a><a NAME="E1:15_1_7_6_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_7_6_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_7_6_1_2"><b>suppose </b></a><a NAME="E1:15_1_7_6_1_2"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <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/><div class="add"><a NAME="E2:15_1_7_6_1_2"/><b>then </b>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c14">a</font> <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="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></i>;<br/><b>hence </b><a NAME="E3:15_1_7_6_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E15:15_1_7"/><b>then </b><i><font color="Green" title="E48">A46</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">X</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<a NAME="E16:15_1_7"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_7_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_7_7"/><i><font color="Green" title="E49">A47</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c17">x</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font> <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">h</font> = <font color="Maroon" title="c18">y</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c14">a</font> <b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1"><i><font color="Green" title="E31">A29</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E4:15_1_7_7"/><i><font color="Green" title="E50">A48</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c'</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E5:15_1_7_7"/><i><font color="Green" title="E51">A49</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/>
<a NAME="E6:15_1_7_7"/><i><font color="Green" title="E52">A50</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</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="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/>
<a NAME="E7:15_1_7_7"/><i><font color="Green" title="E53">A51</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</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="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">tc'</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">tch</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">t1</font> = <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>
<a NAME="E11:15_1_7_7"/><i><font color="Green" title="E54">A52</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</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="yoneda_1.html#E48:15_1"><i><font color="Green" title="E36">A34</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E12:15_1_7_7"/><b>then </b>
<font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<a NAME="E13:15_1_7_7"/><b>then </b><i><font color="Green" title="E55">A53</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">c'</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<i><font color="Green" title="E56">A54</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">t1</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
;<br/>
<a NAME="E15:15_1_7_7"/><i><font color="Green" title="E57">A55</font></i>: 
<font color="Maroon" title="c20">tc'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c20">tc'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></i>;<br/>
<a NAME="E16:15_1_7_7"/><i><font color="Green" title="E58">A56</font></i>: 
<font color="Maroon" title="c21">tch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tch</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></i>;<br/>
<i><font color="Green" title="E59">A57</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
;<br/>
<i><font color="Green" title="E60">A58</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c20">tc'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></i>
;<br/>
<i><font color="Green" title="E61">A59</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tch</font>
<b>by </b><i><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></i>
;<br/>
<i><font color="Green" title="E62">A60</font></i>: <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c19">h</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/>
<i><font color="Green" title="E63">A61</font></i>: <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c19">h</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/>
<a NAME="E22:15_1_7_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c19">h</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E15:15_1_7_7"><i><font color="Green" title="E57">A55</font></i></a>, <a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<a NAME="E23:15_1_7_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c19">h</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<a NAME="E24:15_1_7_7"/><b>then </b><i><font color="Green" title="E64">A62</font></i>: 
<font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c19">h</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>
<i><font color="Green" title="E65">A63</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1_7_7"><i><font color="Green" title="E59">A57</font></i></a>, <a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:15_1_7_7"><i><font color="Green" title="E63">A61</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<i><font color="Green" title="E66">A64</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E18:15_1_7_7"><i><font color="Green" title="E60">A58</font></i></a>, <a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:15_1_7_7"><i><font color="Green" title="E63">A61</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<a NAME="E27:15_1_7_7"/>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c19">h</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tch</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E16:15_1_7_7"><i><font color="Green" title="E58">A56</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1_7_7"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<a NAME="E28:15_1_7_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c19">h</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tch</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tch</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<a NAME="E29:15_1_7_7"/><b>then </b><i><font color="Green" title="E67">A65</font></i>: 
 <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>
<i><font color="Green" title="E68">A66</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E19:15_1_7_7"><i><font color="Green" title="E61">A59</font></i></a>, <a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E20:15_1_7_7"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<i><font color="Green" title="E69">A67</font></i>:  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">t1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7_7"><i><font color="Green" title="E50">A48</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_7_7"><i><font color="Green" title="E55">A53</font></i></a>, <a class="txt" href="yoneda_1.html#E14:15_1_7_7"><i><font color="Green" title="E56">A54</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E20:15_1_7_7"><i><font color="Green" title="E62">A60</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tch</font>
<b>by </b><i><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></i>
;<br/>
<i><font color="Green" title="E70">A68</font></i>:  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c20">tc'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:15_1_7_7"><i><font color="Green" title="E63">A61</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_7"><i><font color="Green" title="E38">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7"><i><font color="Green" title="E39">A37</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_7"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7_7"><i><font color="Green" title="E50">A48</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<a NAME="E33:15_1_7_7"/><i><font color="Green" title="E71">A69</font></i>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_7"><i><font color="Green" title="E38">A36</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_7"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7_7"><i><font color="Green" title="E50">A48</font></i></a>, <a class="txt" href="yoneda_1.html#E26:15_1_7_7"><i><font color="Green" title="E66">A64</font></i></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>;<br/>
<i><font color="Green" title="E72">A70</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1_7"><i><font color="Green" title="E37">A35</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_7_7"><i><font color="Green" title="E52">A50</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">ta1</font>,<font color="Maroon" title="c20">tc'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="yoneda_1.html#E33:15_1_7_7"><i><font color="Green" title="E71">A69</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c15">ta1</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c20">tc'</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E32:15_1_7_7"><i><font color="Green" title="E70">A68</font></i></a>, <a class="ref" href="ens_1.html#D12" title="ENS_1:def.12">ENS_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E32:15_1_7_7"><i><font color="Green" title="E70">A68</font></i></a>, <a class="ref" href="ens_1.html#D7" title="ENS_1:def.7">ENS_1:def 7</a></i>
;<br/>
<a NAME="E35:15_1_7_7"/><i><font color="Green" title="E73">A71</font></i>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7_7"><i><font color="Green" title="E50">A48</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_7_7"><i><font color="Green" title="E55">A53</font></i></a>, <a class="txt" href="yoneda_1.html#E30:15_1_7_7"><i><font color="Green" title="E68">A66</font></i></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>;<br/>
<i><font color="Green" title="E74">A72</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E7:15_1_7_7"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="yoneda_1.html#E11:15_1_7_7"><i><font color="Green" title="E54">A52</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">tch</font>,<font color="Maroon" title="c22">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="yoneda_1.html#E35:15_1_7_7"><i><font color="Green" title="E73">A71</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c21">tch</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c22">t1</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E31:15_1_7_7"><i><font color="Green" title="E69">A67</font></i></a>, <a class="ref" href="ens_1.html#D12" title="ENS_1:def.12">ENS_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">t1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E31:15_1_7_7"><i><font color="Green" title="E69">A67</font></i></a>, <a class="ref" href="ens_1.html#D7" title="ENS_1:def.7">ENS_1:def 7</a></i>
;<br/>
<a NAME="E37:15_1_7_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c'</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></i>;<br/>
<a NAME="E38:15_1_7_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">t1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E34:15_1_7_7"><i><font color="Green" title="E72">A70</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1_7_7"><i><font color="Green" title="E74">A72</font></i></a>, <a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<a NAME="E39:15_1_7_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">t1</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tch</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<a NAME="E40:15_1_7_7"/><b>then </b><i><font color="Green" title="E75">A73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>
<a NAME="E41:15_1_7_7"/>
<font color="Maroon" title="c20">tc'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c20">tc'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T19" title="ENS_1:th.19">ENS_1:19</a></i>;<br/>
<a NAME="E42:15_1_7_7"/><b>then </b><i><font color="Green" title="E76">A74</font></i>: 
<font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c20">tc'</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c20">tc'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T20" title="ENS_1:th.20">ENS_1:20</a></i>;<br/>
<a NAME="E43:15_1_7_7"/><i><font color="Green" title="E77">A75</font></i>: 
<font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">h</font></span>)</span>, <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c19">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1_7_7"><i><font color="Green" title="E59">A57</font></i></a>, <a class="txt" href="yoneda_1.html#E18:15_1_7_7"><i><font color="Green" title="E60">A58</font></i></a>, <a class="txt" href="yoneda_1.html#E25:15_1_7_7"><i><font color="Green" title="E65">A63</font></i></a>, <a class="txt" href="yoneda_1.html#E26:15_1_7_7"><i><font color="Green" title="E66">A64</font></i></a>, <a class="txt" href="yoneda_1.html#E42:15_1_7_7"><i><font color="Green" title="E76">A74</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E44:15_1_7_7"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c4">c'</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_7_7"><i><font color="Green" title="E50">A48</font></i></a>, <a class="txt" href="yoneda_1.html#E43:15_1_7_7"><i><font color="Green" title="E77">A75</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E45:15_1_7_7"/><b>then </b><i><font color="Green" title="E78">A76</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/>
<a NAME="E46:15_1_7_7"/><i><font color="Green" title="E79">A77</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E42:15_1"><i><font color="Green" title="E34">A32</font></i></a>, <a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/>
<a NAME="E47:15_1_7_7"/><i><font color="Green" title="E80">A78</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<a NAME="E48:15_1_7_7"/><i><font color="Green" title="E81">A79</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c'</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/>
<span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<b>by </b><i><a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></i>
<br/>.= 
<font color="Maroon" title="c18">y</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c13">f</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_7"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_7_7"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="ens_1.html#D21" title="ENS_1:def.21">ENS_1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_7_7"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="yoneda_1.html#E29:15_1_7_7"><i><font color="Green" title="E67">A65</font></i></a>, <a class="txt" href="yoneda_1.html#E47:15_1_7_7"><i><font color="Green" title="E80">A78</font></i></a>, <a class="ref" href="ens_1.html#D20" title="ENS_1:def.20">ENS_1:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">tc'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E40:15_1_7_7"><i><font color="Green" title="E75">A73</font></i></a>, <a class="txt" href="yoneda_1.html#E46:15_1_7_7"><i><font color="Green" title="E79">A77</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c'</font>,<font color="Maroon" title="c19">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E24:15_1_7_7"><i><font color="Green" title="E64">A62</font></i></a>, <a class="txt" href="yoneda_1.html#E45:15_1_7_7"><i><font color="Green" title="E78">A76</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">y</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_7_7"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="yoneda_1.html#E48:15_1_7_7"><i><font color="Green" title="E81">A79</font></i></a>, <a class="ref" href="ens_1.html#D20" title="ENS_1:def.20">ENS_1:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="yoneda_1.html#E5:15_1_7_7"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>
;<br/>
<b>hence </b><a NAME="E50:15_1_7_7"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:15_1_7"/><b>then </b>
 <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E15:15_1_7"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E18:15_1_7"/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c13">f</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_7"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1_7"><i><font color="Green" title="E42">A40</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1_7"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:15_1"/><b>then </b>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c13">f</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">t</font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="isocat_1.html#T31" title="ISOCAT_1:th.31">ISOCAT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E51:15_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
 <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font> is <a href="yoneda_1.html#V2" title="YONEDA_1:attr.2">full</a>
 <b>by </b><i><a class="ref" href="yoneda_1.html#D7" target="_self" title="YONEDA_1:def.7">Def7</a></i>;<br/>


</div>
