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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:15"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ex <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E11</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E2:15_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:15_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E5:15_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16">NATTRA_1:def 16</a></i>;<br/>
<a NAME="E6:15_1"/><i><font color="Green">E13</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="nattra_1.html#T39">NATTRA_1:39</a></i>;<br/>
<a NAME="E7:15_1"/><i><font color="Green">E14</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E8:15_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E9:15_1"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> as   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E11:15_1"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font> =  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>;<br/>
<i><font color="Green">E16</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> = 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E2:15_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:15_1_1_2"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E4:15_1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E5:15_1_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font> ;<br/>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T15">NATTRA_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#D5">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" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:15_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#D5">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">E17</font></i>: 
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="nattra_1.html#K7">id</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E3:15_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#D18">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" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:15_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T31">OPPCAT_1:31</a></i>;<br/>
<a NAME="E13:15_1"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> as   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E15:15_1"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></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">c<sub>11</sub></font> =  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>;<br/>
<i><font color="Green">E17</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> = 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E2:15_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:15_1_2_2"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E4:15_1_2_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<a NAME="E5:15_1_2_2"/><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font> ;<br/>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_2_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T15">NATTRA_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#D5">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" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E4:15_1_2_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="nattra_1.html#D5">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">E18</font></i>: 
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="nattra_1.html#K7">id</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="yoneda_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E3:15_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="nattra_1.html#D18">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" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:15_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T31">OPPCAT_1:31</a></i>;<br/>
<i><font color="Green">E18</font></i>: <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#T60">ENS_1:60</a></i>
;<br/>
<i><font color="Green">E19</font></i>: <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#T60">ENS_1:60</a></i>
;<br/>
<a NAME="E19:15_1"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E16:15_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E20:15_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yoneda_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E12:15_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E21:15_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <b>by </b><i><a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yoneda_1.html#E16:15_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E23:15_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:15_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E24:15_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E25:15_1"/><b>then </b><i><font color="Green">E24</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<i><font color="Green">E25</font></i>:  <a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="ens_1.html#K12">Ens</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
 <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> #)
<b>by </b><i><a class="ref" href="ens_1.html#D14">ENS_1:def 14</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E28:15_1"/>
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E18:15_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<a NAME="E29:15_1"/><b>then </b>
the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>;<br/>
<a NAME="E30:15_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>;<br/>
<a NAME="E31:15_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T19">ENS_1:19</a></i>;<br/>
<a NAME="E32:15_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T20">ENS_1:20</a></i>;<br/>
<a NAME="E33:15_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E30:15_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#T14">FUNCT_2:14</a></i>;<br/>
<a NAME="E34:15_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>;<br/>
<a NAME="E35:15_1"/><i><font color="Green">E30</font></i>: 
the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yoneda_1.html#E33:15_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>;<br/>
<a NAME="E36:15_1"/><i><font color="Green">E31</font></i>: 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yoneda_1.html#E34:15_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>;<br/>
<a NAME="E37:15_1"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yoneda_1.html#E35:15_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E38:15_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yoneda_1.html#E24:15_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>

<i><font color="Green">E33</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
;<br/>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
;<br/>
<a NAME="E41:15_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yoneda_1.html#E32:15_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="yoneda_1.html#E39:15_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E42:15_1"/><b>then </b><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yoneda_1.html#E18:15_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E43:15_1"/><b>then </b>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<a NAME="E44:15_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E45:15_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E42:15_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<a NAME="E47:15_1"/><i><font color="Green">E35</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yoneda_1.html#E37:15_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E48:15_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<a NAME="E49:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:15_1_8"/><i><font color="Green">E37</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E48:15_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E2:15_1_8"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<i><font color="Green">E39</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
;<br/>
<i><font color="Green">E40</font></i>:  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
;<br/>
<i><font color="Green">E41</font></i>: <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#T60">ENS_1:60</a></i>
;<br/>
<i><font color="Green">E42</font></i>: <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D26">ENS_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D20">CAT_1:def 20</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#T60">ENS_1:60</a></i>
;<br/>
<a NAME="E8:15_1_8"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_8"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8"><i><font color="Green">E39</font></i></a>, <a class="txt" href="yoneda_1.html#E5:15_1_8"><i><font color="Green">E40</font></i></a>, <a class="ref" href="ens_1.html#T8">ENS_1:8</a></i>;<br/>
<a NAME="E9:15_1_8"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T19">ENS_1:19</a></i>;<br/>
<a NAME="E10:15_1_8"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T20">ENS_1:20</a></i>;<br/>
<a NAME="E11:15_1_8"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>15</sub></font>, <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E12:15_1_8"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>, <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E4:15_1_8"><i><font color="Green">E39</font></i></a>, <a class="txt" href="yoneda_1.html#E5:15_1_8"><i><font color="Green">E40</font></i></a>, <a class="txt" href="yoneda_1.html#E10:15_1_8"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E13:15_1_8"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_8_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_8_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_8_5_1"/>
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_8_5_1_1"><b>suppose </b></a><a NAME="E1:15_1_8_5_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_8_5_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_8"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8"><i><font color="Green">E39</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_8"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yoneda_1.html#E11:15_1_8"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_8_5_1_2"><b>suppose </b></a><a NAME="E1:15_1_8_5_1_2"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:15_1_8_5_1_2"/><b>then </b>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ens_1.html#T42">ENS_1:42</a></i>;<br/><b>hence </b><a NAME="E3:15_1_8_5_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yoneda_1.html#E2:15_1_8"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_8"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1_8"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yoneda_1.html#E12:15_1_8"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#D1">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_8"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_8_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_8_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_8_6_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_8_6_1_1"><b>suppose </b></a><a NAME="E1:15_1_8_6_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_8_6_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_8_6_1_2"><b>suppose </b></a><a NAME="E1:15_1_8_6_1_2"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:15_1_8_6_1_2"/><b>then </b>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ens_1.html#T42">ENS_1:42</a></i>;<br/><b>hence </b><a NAME="E3:15_1_8_6_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#D1">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_8"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>;<br/>
<a NAME="E16:15_1_8"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_8_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_8_7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yoneda_1.html#E25:15_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E4:15_1_8_7"/><i><font color="Green">E50</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E5:15_1_8_7"/><i><font color="Green">E51</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E6:15_1_8_7"/><i><font color="Green">E52</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/>
<a NAME="E7:15_1_8_7"/><i><font color="Green">E53</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E11:15_1_8_7"/><i><font color="Green">E54</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E48:15_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E12:15_1_8_7"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E13:15_1_8_7"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<i><font color="Green">E56</font></i>:  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
;<br/>
<a NAME="E15:15_1_8_7"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8">ENS_1:8</a></i>;<br/>
<a NAME="E16:15_1_8_7"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8">ENS_1:8</a></i>;<br/>
<i><font color="Green">E59</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
;<br/>
<i><font color="Green">E60</font></i>:  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D11">ENS_1:def 11</a></i>
;<br/>
<i><font color="Green">E61</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D10">ENS_1:def 10</a></i>
;<br/>
<i><font color="Green">E62</font></i>: <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/>
<i><font color="Green">E63</font></i>: <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D22">ENS_1:def 22</a></i>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/>
<a NAME="E22:15_1_8_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E15:15_1_8_7"><i><font color="Green">E57</font></i></a>, <a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E23:15_1_8_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E24:15_1_8_7"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<i><font color="Green">E65</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1_8_7"><i><font color="Green">E59</font></i></a>, <a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:15_1_8_7"><i><font color="Green">E63</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<i><font color="Green">E66</font></i>:  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E18:15_1_8_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:15_1_8_7"><i><font color="Green">E63</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<a NAME="E27:15_1_8_7"/>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E16:15_1_8_7"><i><font color="Green">E58</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1_8_7"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E28:15_1_8_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E29:15_1_8_7"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<i><font color="Green">E68</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E19:15_1_8_7"><i><font color="Green">E61</font></i></a>, <a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E20:15_1_8_7"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<i><font color="Green">E69</font></i>:  <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8_7"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_8_7"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yoneda_1.html#E14:15_1_8_7"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E20:15_1_8_7"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
;<br/>
<i><font color="Green">E70</font></i>:  <a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="yoneda_1.html#E21:15_1_8_7"><i><font color="Green">E63</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_8"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8"><i><font color="Green">E39</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_8"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8_7"><i><font color="Green">E50</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<a NAME="E33:15_1_8_7"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E2:15_1_8"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_8"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8_7"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yoneda_1.html#E26:15_1_8_7"><i><font color="Green">E66</font></i></a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>
<i><font color="Green">E72</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E1:15_1_8"><i><font color="Green">E37</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_8_7"><i><font color="Green">E52</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yoneda_1.html#E33:15_1_8_7"><i><font color="Green">E71</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="ens_1.html#K6">*</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E32:15_1_8_7"><i><font color="Green">E70</font></i></a>, <a class="ref" href="ens_1.html#D12">ENS_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E32:15_1_8_7"><i><font color="Green">E70</font></i></a>, <a class="ref" href="ens_1.html#D7">ENS_1:def 7</a></i>
;<br/>
<a NAME="E35:15_1_8_7"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8_7"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_8_7"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yoneda_1.html#E30:15_1_8_7"><i><font color="Green">E68</font></i></a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>
<i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E7:15_1_8_7"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yoneda_1.html#E11:15_1_8_7"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E26:15_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yoneda_1.html#E35:15_1_8_7"><i><font color="Green">E73</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="ens_1.html#K6">*</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E31:15_1_8_7"><i><font color="Green">E69</font></i></a>, <a class="ref" href="ens_1.html#D12">ENS_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E31:15_1_8_7"><i><font color="Green">E69</font></i></a>, <a class="ref" href="ens_1.html#D7">ENS_1:def 7</a></i>
;<br/>
<a NAME="E37:15_1_8_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="yoneda_1.html#K2" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>;<br/>
<a NAME="E38:15_1_8_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E34:15_1_8_7"><i><font color="Green">E72</font></i></a>, <a class="txt" href="yoneda_1.html#E36:15_1_8_7"><i><font color="Green">E74</font></i></a>, <a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E39:15_1_8_7"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3">MCART_1:def 3</a></i>;<br/>
<a NAME="E40:15_1_8_7"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<a NAME="E41:15_1_8_7"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T19">ENS_1:19</a></i>;<br/>
<a NAME="E42:15_1_8_7"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ens_1.html#T20">ENS_1:20</a></i>;<br/>
<a NAME="E43:15_1_8_7"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>, <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E17:15_1_8_7"><i><font color="Green">E59</font></i></a>, <a class="txt" href="yoneda_1.html#E18:15_1_8_7"><i><font color="Green">E60</font></i></a>, <a class="txt" href="yoneda_1.html#E25:15_1_8_7"><i><font color="Green">E65</font></i></a>, <a class="txt" href="yoneda_1.html#E26:15_1_8_7"><i><font color="Green">E66</font></i></a>, <a class="txt" href="yoneda_1.html#E42:15_1_8_7"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E44:15_1_8_7"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yoneda_1.html#E4:15_1_8_7"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yoneda_1.html#E43:15_1_8_7"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E45:15_1_8_7"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<a NAME="E46:15_1_8_7"/><i><font color="Green">E79</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E42:15_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<a NAME="E47:15_1_8_7"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E48:15_1_8_7"/><i><font color="Green">E81</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<span class="p1">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K19">hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="ens_1.html#T53">ENS_1:53</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yoneda_1.html#E13:15_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yoneda_1.html#E1:15_1_8_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ens_1.html#D21">ENS_1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_8_7"><i><font color="Green">E51</font></i></a>, <a class="txt" href="yoneda_1.html#E29:15_1_8_7"><i><font color="Green">E67</font></i></a>, <a class="txt" href="yoneda_1.html#E47:15_1_8_7"><i><font color="Green">E80</font></i></a>, <a class="ref" href="ens_1.html#D20">ENS_1:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E40:15_1_8_7"><i><font color="Green">E75</font></i></a>, <a class="txt" href="yoneda_1.html#E46:15_1_8_7"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18">hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E24:15_1_8_7"><i><font color="Green">E64</font></i></a>, <a class="txt" href="yoneda_1.html#E45:15_1_8_7"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1_8_7"><i><font color="Green">E51</font></i></a>, <a class="txt" href="yoneda_1.html#E48:15_1_8_7"><i><font color="Green">E81</font></i></a>, <a class="ref" href="ens_1.html#D20">ENS_1:def 20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yoneda_1.html#E5:15_1_8_7"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>
;<br/>
<b>hence </b><a NAME="E50:15_1_8_7"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K22">hom</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:15_1_8"/><b>then </b>
 <a href="ens_1.html#K22">hom</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E15:15_1_8"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E18:15_1_8"/>
<span class="p1"><a href="yoneda_1.html#K3" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1_8"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1_8"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1_8"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yoneda_1.html#D3" target="_self">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" target="_self">&lt;|</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="yoneda_1.html#K3" target="_self">,?&gt;</a></span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yoneda_1.html#E8:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="isocat_1.html#T31">ISOCAT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E51:15_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="yoneda_1.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yoneda_1.html#E6:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yoneda_1.html#E7:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yoneda_1.html#E19:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yoneda_1.html#E20:15_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yoneda_1.html#E47:15_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yoneda_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
 <a href="yoneda_1.html#K4" target="_self">Yoneda</a> <font color="Maroon">c<sub>1</sub></font> is <a href="yoneda_1.html#V2" target="_self">full</a>
 <b>by </b><i><a class="ref" href="yoneda_1.html#D7" target="_self">Def7</a></i>;<br/>


</div>
