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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font> is  <a href="yoneda_1.html#V2" title="YONEDA_1:attr.2">full</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">F</font> =  <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E1:16"/>
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">c9</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b2">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b2">c9</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Olive" title="b1">c</font> holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b4">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c9</font> st <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">c9</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font> holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font> holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> )</span><br/>

<span class="lab"><font color="Green" title="E11">A2</font></span>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c9</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c9</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c4">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> st <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )</span><br/>

<a NAME="E3:16_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>, <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c8">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font><span class="kw"> such that </span><br/><a NAME="E6:16_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:16_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></span>;<br/>
<a NAME="E8:16_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></span>;<br/>
<a NAME="E9:16_1"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">d1</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></span>;<br/>
<a NAME="E11:16_1"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">d</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></span>;<br/>
<a NAME="E13:16_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c10">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>;<br/>
<a NAME="E1:16_1_2"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> implies <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">z</font> = <font color="Maroon" title="c12">y</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">zz</font> =  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> ;<br/>
<a NAME="E4:16_1_2_1"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></span>;<br/>
<a NAME="E5:16_1_2_1"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<a NAME="E6:16_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">z</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></span>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">zz</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_2_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T15" title="NATTRA_1:th.15">NATTRA_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T21" title="NATTRA_1:th.21">NATTRA_1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:16_1_2_1"/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_2_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E3:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c10">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:16_1_2"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c10">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T31" title="OPPCAT_1:th.31">OPPCAT_1:31</a></span>;<br/>
<a NAME="E15:16_1"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">d1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>;<br/>
<a NAME="E1:16_1_3"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> holds <br/><span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> implies <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_3_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">z</font> = <font color="Maroon" title="c12">y</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">zz</font> =  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> ;<br/>
<a NAME="E4:16_1_3_1"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></span>;<br/>
<a NAME="E5:16_1_3_1"/>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<a NAME="E6:16_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">z</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c13">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c13">z</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c13">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></span>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">zz</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_3_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T15" title="NATTRA_1:th.15">NATTRA_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T21" title="NATTRA_1:th.21">NATTRA_1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:16_1_3_1"/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_3_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E3:16_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A16</font></span>: 
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_3"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_3"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">d1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:16_1_3"/>
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">d1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A17</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T31" title="OPPCAT_1:th.31">OPPCAT_1:31</a></span>;<br/>
<a NAME="E17:16_1"/><span class="lab"><font color="Green" title="E19">A18</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></span>;<br/>
<a NAME="E18:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A19</font></span>: 
<font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<a NAME="E19:16_1"/><span class="lab"><font color="Green" title="E21">A20</font></span>: 
<span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="oppcat_1.html#T31" title="OPPCAT_1:th.31">OPPCAT_1:31</a></span>;<br/>
<span class="lab"><font color="Green" title="E22">A21</font></span>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></span>
;<br/>
<a NAME="E21:16_1"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c3">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c8">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>,<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E17:16_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E15:16_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E14:16_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="oppcat_1.html#T31" title="OPPCAT_1:th.31">OPPCAT_1:31</a></span>;<br/>
<a NAME="E23:16_1"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="oppcat_1.html#T31" title="OPPCAT_1:th.31">OPPCAT_1:31</a></span>;<br/>
<a NAME="E24:16_1"/><span class="kw">then </span>
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E17:16_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E21:16_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="txt" href="#E19:16_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></span>;<br/>
<a NAME="E25:16_1"/><span class="kw">then </span>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></span>;<br/>
<a NAME="E26:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A23</font></span>: 
<font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<a NAME="E27:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A24</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E28:16_1"/><span class="lab"><font color="Green" title="E26">A25</font></span>: 
 <a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D14" title="ENS_1:def.14">ENS_1:def 14</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">t1</font> = <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E27">A26</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></span>
;<br/>
<a NAME="E31:16_1"/>
<font color="Maroon" title="c12">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T19" title="ENS_1:th.19">ENS_1:19</a></span>;<br/>
<a NAME="E32:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A27</font></span>: 
<font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T20" title="ENS_1:th.20">ENS_1:20</a></span>;<br/>
<a NAME="E33:16_1"/><span class="lab"><font color="Green" title="E29">A28</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16_1"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E34:16_1"/><span class="kw">then </span>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></span>;<br/>
<a NAME="E35:16_1"/><span class="kw">then </span>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></span>;<br/>
<a NAME="E36:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A29</font></span>: 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:16_1"><span class="lab"><font color="Green" title="E28">A27</font></span></a></span>;<br/>
<a NAME="E37:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A30</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:16_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E26:16_1"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> st <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span><br/>

 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">t1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></span>
;<br/>
<a NAME="E40:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:16_1"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="txt" href="#E27:16_1"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E32:16_1"><span class="lab"><font color="Green" title="E28">A27</font></span></a>, <a class="txt" href="#E30:16_1"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/>
<a NAME="E41:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A32</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E27:16_1"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E37:16_1"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E42:16_1"/><span class="kw">then </span>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></span>;<br/>
<a NAME="E43:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E44:16_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E40:16_1"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E45:16_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:16_1"><span class="lab"><font color="Green" title="E34">A33</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">c9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<a NAME="E47:16_1"/><span class="lab"><font color="Green" title="E35">A34</font></span>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:16_1"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E36:16_1"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></span>;<br/>
<a NAME="E48:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A35</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span><br/>

<a NAME="E49:16_1"/><span class="lab"><font color="Green" title="E37">A36</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E47:16_1"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<a NAME="E50:16_1"/><span class="lab"><font color="Green" title="E38">A37</font></span>: 
<font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yoneda_1.html#K5" title="YONEDA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<a NAME="E51:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A38</font></span>: 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E18:16_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E16:16_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E14:16_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></span>;<br/>
<a NAME="E52:16_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c13">f</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c13">f</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span><br/>

<span class="lab"><font color="Green" title="E40">A39</font></span>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c14">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">ta1</font> = <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E41">A40</font></span>: <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c9</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D26" title="ENS_1:def.26">ENS_1:def 26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><a href="ens_1.html#K24" title="ENS_1:func.24">hom?-</a> <span class="p3">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">c9</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T60" title="ENS_1:th.60">ENS_1:60</a></span>
;<br/>
<a NAME="E4:16_1_7"/>
<font color="Maroon" title="c15">ta1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T19" title="ENS_1:th.19">ENS_1:19</a></span>;<br/>
<a NAME="E5:16_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A41</font></span>: 
<font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T20" title="ENS_1:th.20">ENS_1:20</a></span>;<br/>
<a NAME="E6:16_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A42</font></span>: 
<font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<span class="lab"><font color="Green" title="E44">A43</font></span>:  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></span>
;<br/>
<a NAME="E8:16_1_7"/><span class="lab"><font color="Green" title="E45">A44</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:16_1"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></span>;<br/>
<a NAME="E9:16_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A45</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<a NAME="E10:16_1_7"/><span class="lab"><font color="Green" title="E47">A46</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_7"><span class="lab"><font color="Green" title="E45">A44</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<span class="lab"><font color="Green" title="E48">A47</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></span>
;<br/>
<a NAME="E12:16_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A48</font></span>: 
<font color="Maroon" title="c15">ta1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_7"><span class="lab"><font color="Green" title="E46">A45</font></span></a>, <a class="txt" href="#E10:16_1_7"><span class="lab"><font color="Green" title="E47">A46</font></span></a>, <a class="txt" href="#E7:16_1_7"><span class="lab"><font color="Green" title="E44">A43</font></span></a>, <a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></span>;<br/>
<a NAME="E13:16_1_7"/><span class="lab"><font color="Green" title="E50">A49</font></span>: 
<font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_7"><span class="lab"><font color="Green" title="E44">A43</font></span></a>, <a class="txt" href="#E11:16_1_7"><span class="lab"><font color="Green" title="E48">A47</font></span></a>, <a class="txt" href="#E5:16_1_7"><span class="lab"><font color="Green" title="E42">A41</font></span></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/>
<a NAME="E14:16_1_7"/><span class="lab"><font color="Green" title="E51">A50</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_7_5"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_7_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_7_5_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_7_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_7_5_1_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_1_7_5_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_7"><span class="lab"><font color="Green" title="E46">A45</font></span></a>, <a class="txt" href="#E7:16_1_7"><span class="lab"><font color="Green" title="E44">A43</font></span></a>, <a class="txt" href="#E3:16_1_7"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E6:16_1_7"><span class="lab"><font color="Green" title="E43">A42</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_7_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_7_5_1_2"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font></span><br/><div class="add"><a NAME="E2:16_1_7_5_1_2"/><span class="kw">then </span>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_7_5_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:16_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E9:16_1_7"><span class="lab"><font color="Green" title="E46">A45</font></span></a>, <a class="txt" href="#E10:16_1_7"><span class="lab"><font color="Green" title="E47">A46</font></span></a>, <a class="txt" href="#E3:16_1_7"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E1:16_1_7"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="txt" href="#E13:16_1_7"><span class="lab"><font color="Green" title="E50">A49</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E15:16_1_7"/><span class="lab"><font color="Green" title="E52">A51</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_7_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c17">t1</font> = <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> implies <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )</span><br/>

<a NAME="E2:16_1_7_6"/><span class="lab"><font color="Green" title="E53">A52</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:16_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16_1_7_6"/><span class="lab"><font color="Green" title="E54">A53</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c18">x</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">h</font> = <font color="Maroon" title="c19">y</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c14">a</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E27:16_1"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E37:16_1"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<a NAME="E6:16_1_7_6"/><span class="lab"><font color="Green" title="E55">A54</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">tc9</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>;<br/>
<a NAME="E8:16_1_7_6"/><span class="lab"><font color="Green" title="E56">A55</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">tch</font> = <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A56</font></span>: <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c20">h</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></span>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
;<br/>
<span class="lab"><font color="Green" title="E58">A57</font></span>: <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c20">h</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D22" title="ENS_1:def.22">ENS_1:def 22</a></span>
<br/>.= 
<span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
;<br/>
<span class="lab"><font color="Green" title="E59">A58</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tc9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E60">A59</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_7_6"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<a NAME="E14:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A60</font></span>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_7"><span class="lab"><font color="Green" title="E46">A45</font></span></a>, <a class="txt" href="#E3:16_1_7"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E8:16_1_7_6"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></span>;<br/>
<a NAME="E15:16_1_7_6"/>
<font color="Maroon" title="c21">tc9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tc9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T19" title="ENS_1:th.19">ENS_1:19</a></span>;<br/>
<a NAME="E16:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A61</font></span>: 
<font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tc9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T20" title="ENS_1:th.20">ENS_1:20</a></span>;<br/>
<span class="lab"><font color="Green" title="E63">A62</font></span>:  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></span>
;<br/>
<span class="kw">then </span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_7_6"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<a NAME="E19:16_1_7_6"/><span class="kw">then </span>
<font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16_1_7_6"><span class="lab"><font color="Green" title="E63">A62</font></span></a>, <a class="txt" href="#E12:16_1_7_6"><span class="lab"><font color="Green" title="E59">A58</font></span></a>, <a class="txt" href="#E13:16_1_7_6"><span class="lab"><font color="Green" title="E60">A59</font></span></a>, <a class="txt" href="#E16:16_1_7_6"><span class="lab"><font color="Green" title="E62">A61</font></span></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/>
<a NAME="E20:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A63</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="txt" href="#E6:16_1_7_6"><span class="lab"><font color="Green" title="E55">A54</font></span></a>, <a class="txt" href="#E8:16_1_7_6"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E21:16_1_7_6"/><span class="lab"><font color="Green" title="E65">A64</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>;<br/>
<a NAME="E22:16_1_7_6"/>
<font color="Maroon" title="c22">tch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">tch</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></span>;<br/>
<a NAME="E23:16_1_7_6"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c20">h</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">tch</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_7_6"><span class="lab"><font color="Green" title="E57">A56</font></span></a>, <a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E24:16_1_7_6"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c20">h</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">tch</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">tch</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E25:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A65</font></span>: 
 <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></span>;<br/>
<a NAME="E26:16_1_7_6"/>
<font color="Maroon" title="c21">tc9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tc9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T8" title="ENS_1:th.8">ENS_1:8</a></span>;<br/>
<a NAME="E27:16_1_7_6"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c20">h</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_7_6"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E28:16_1_7_6"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c20">h</font></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E29:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A66</font></span>: 
<font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c20">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></span>;<br/>
<a NAME="E30:16_1_7_6"/><span class="lab"><font color="Green" title="E68">A67</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:16_1"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></span>;<br/>
<a NAME="E31:16_1_7_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<a NAME="E32:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A68</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">tch</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D10" title="ENS_1:def.10">ENS_1:def 10</a></span>
;<br/>
<span class="kw">then </span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_7_6"><span class="lab"><font color="Green" title="E57">A56</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<a NAME="E35:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A69</font></span>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E6:16_1_7_6"><span class="lab"><font color="Green" title="E55">A54</font></span></a>, <a class="txt" href="#E32:16_1_7_6"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></span>;<br/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c17">t1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" title="ENS_1:def.11">ENS_1:def 11</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E71">A70</font></span>:  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c17">t1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E6:16_1_7_6"><span class="lab"><font color="Green" title="E55">A54</font></span></a>, <a class="txt" href="#E32:16_1_7_6"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_7_6"><span class="lab"><font color="Green" title="E57">A56</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c22">tch</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D4" title="ENS_1:def.4">ENS_1:def 4</a></span>
;<br/>
<a NAME="E38:16_1_7_6"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E72">A71</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:16_1_7_6"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">tch</font>,<font color="Maroon" title="c17">t1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E35:16_1_7_6"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c22">tch</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c17">t1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:16_1_7_6"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="ens_1.html#D12" title="ENS_1:def.12">ENS_1:def 12</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c17">t1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:16_1_7_6"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="ens_1.html#D7" title="ENS_1:def.7">ENS_1:def 7</a></span>
;<br/>
<span class="lab"><font color="Green" title="E73">A72</font></span>:  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c21">tc9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D5" title="ENS_1:def.5">ENS_1:def 5</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_7_6"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c15">ta1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_7"><span class="lab"><font color="Green" title="E46">A45</font></span></a>, <a class="txt" href="#E7:16_1_7"><span class="lab"><font color="Green" title="E44">A43</font></span></a>, <a class="txt" href="#E3:16_1_7"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E8:16_1_7_6"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<a NAME="E41:16_1_7_6"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E74">A73</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_7"><span class="lab"><font color="Green" title="E45">A44</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17" title="ENS_1:func.17">Hom</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">ta1</font>,<font color="Maroon" title="c21">tc9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E14:16_1_7_6"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c15">ta1</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c21">tc9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:16_1_7_6"><span class="lab"><font color="Green" title="E73">A72</font></span></a>, <a class="ref" href="ens_1.html#D12" title="ENS_1:def.12">ENS_1:def 12</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:16_1_7_6"><span class="lab"><font color="Green" title="E73">A72</font></span></a>, <a class="ref" href="ens_1.html#D7" title="ENS_1:def.7">ENS_1:def 7</a></span>
;<br/>
<a NAME="E43:16_1_7_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c4">c9</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">&lt;|</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="yoneda_1.html#K2" title="YONEDA_1:func.2">,?&gt;</a></span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E50:16_1"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="txt" href="#E18:16_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E16:16_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E14:16_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></span>;<br/>
<a NAME="E44:16_1_7_6"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c17">t1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">tch</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16_1_7_6"><span class="lab"><font color="Green" title="E74">A73</font></span></a>, <a class="txt" href="#E39:16_1_7_6"><span class="lab"><font color="Green" title="E72">A71</font></span></a>, <a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E45:16_1_7_6"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c21">tc9</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c15">ta1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c17">t1</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c22">tch</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></span>;<br/>
<a NAME="E46:16_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A74</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></span>;<br/>
<a NAME="E47:16_1_7_6"/><span class="lab"><font color="Green" title="E76">A75</font></span>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">c9</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></span>;<br/>
<span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></span>
<br/>.= 
<font color="Maroon" title="c19">y</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c13">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E3:16_1_7_6"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="ens_1.html#D21" title="ENS_1:def.21">ENS_1:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">tch</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_1_7_6"><span class="lab"><font color="Green" title="E65">A64</font></span></a>, <a class="txt" href="#E25:16_1_7_6"><span class="lab"><font color="Green" title="E66">A65</font></span></a>, <a class="txt" href="#E2:16_1_7_6"><span class="lab"><font color="Green" title="E53">A52</font></span></a>, <a class="ref" href="ens_1.html#D20" title="ENS_1:def.20">ENS_1:def 20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">tc9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:16_1"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="txt" href="#E46:16_1_7_6"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" title="ENS_1:func.18">hom</a> <font color="Maroon" title="c4">c9</font>,<font color="Maroon" title="c20">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:16_1_7_6"><span class="lab"><font color="Green" title="E67">A66</font></span></a>, <a class="txt" href="#E20:16_1_7_6"><span class="lab"><font color="Green" title="E64">A63</font></span></a>, <a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">c9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_1_7_6"><span class="lab"><font color="Green" title="E65">A64</font></span></a>, <a class="txt" href="#E47:16_1_7_6"><span class="lab"><font color="Green" title="E76">A75</font></span></a>, <a class="ref" href="ens_1.html#D20" title="ENS_1:def.20">ENS_1:def 20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E21:16_1_7_6"><span class="lab"><font color="Green" title="E65">A64</font></span></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E49:16_1_7_6"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:16_1_7"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_7_7"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_7_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_7_7_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_7_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_7_7_1_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_1_7_7_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_7_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_7_7_1_2"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font></span><br/><div class="add"><a NAME="E2:16_1_7_7_1_2"/><span class="kw">then </span>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:16_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="ens_1.html#T42" title="ENS_1:th.42">ENS_1:42</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_7_7_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" title="ENS_1:func.19">hom</a> <font color="Maroon" title="c13">f</font>,<font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font></span>)</span>,<font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:16_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E17:16_1_7"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E14:16_1_7"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="ref" href="ens_1.html#T53" title="ENS_1:th.53">ENS_1:53</a></span>;<br/>
<a NAME="E18:16_1_7"/><span class="kw">then </span>
 <a href="ens_1.html#K22" title="ENS_1:func.22">hom</a> <font color="Maroon" title="c13">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ta1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_1_7"><span class="lab"><font color="Green" title="E52">A51</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:16_1_7"/>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c13">f</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:16_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E3:16_1_7"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E1:16_1_7"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="txt" href="#E12:16_1_7"><span class="lab"><font color="Green" title="E49">A48</font></span></a>, <a class="ref" href="yoneda_1.html#D3" target="_self" title="YONEDA_1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E53:16_1"/><span class="kw">then </span>
<span class="p1"><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">&lt;|</a><span class="default"><font color="Maroon" title="c13">f</font></span><a href="yoneda_1.html#K3" title="YONEDA_1:func.3">,?&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E50:16_1"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="txt" href="#E18:16_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E16:16_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E14:16_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E48:16_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="isocat_1.html#T31" title="ISOCAT_1:th.31">ISOCAT_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E54:16_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E17:16_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E21:16_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="txt" href="#E16:16_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E14:16_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E48:16_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E49:16_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="yoneda_1.html#D4" target="_self" title="YONEDA_1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:16"/>
 <a href="yoneda_1.html#K4" title="YONEDA_1:func.4">Yoneda</a> <font color="Maroon" title="c1">A</font> is  <a href="yoneda_1.html#V2" title="YONEDA_1:attr.2">full</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yoneda_1.html#D7" target="_self" title="YONEDA_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
