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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">m1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">m2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:2"/><i><font color="Green">E17</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m1</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> )
 ;<br/>

<a NAME="E2:2"/><i><font color="Green">E18</font></i>: 
 <a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K2">Maps</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8">fDom</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9">fCod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11">fId</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ens_1.html#D14">ENS_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m1'</font> = <font color="Maroon">m1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">m2'</font> = <font color="Maroon">m2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:2"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="yoneda_1.html#K1" target="_self">EnsHom</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>
<i><font color="Green">E22</font></i>:  <a href="ens_1.html#K4">cod</a> <font color="Maroon">m1'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m1</font></span>)</span></span><a href="cat_2.html#K12">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">m2</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font></span>)</span></span><a href="cat_2.html#K12">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K3">dom</a> <font color="Maroon">m2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
;<br/>
<a NAME="E7:2"/><i><font color="Green">E23</font></i>: 
(  <a href="ens_1.html#K3">dom</a> <font color="Maroon">m1'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font> &amp;  <a href="ens_1.html#K4">cod</a> <font color="Maroon">m2'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<i><font color="Green">E24</font></i>:  <a href="ens_1.html#K3">dom</a> <font color="Maroon">m1'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="ens_1.html#D4">ENS_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m1</font></span>)</span></span><a href="cat_2.html#K12">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font> = 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font></span>)</span></span><a href="cat_2.html#K12">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="ens_1.html#K4">cod</a> <font color="Maroon">m2'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D5">ENS_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E3:2_2"/>
(  <a href="ens_1.html#K3">dom</a> <font color="Maroon">m1'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font> &amp;  <a href="ens_1.html#K4">cod</a> <font color="Maroon">m2'</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">m2'</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">m1'</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

<a NAME="E1:2_3"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">m2'</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">m2'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">m2'</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8">ENS_1:8</a></i>;<br/>
<a NAME="E2:2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3">dom</a> <font color="Maroon">m1'</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4">cod</a> <font color="Maroon">m1'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">m1'</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T8">ENS_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:2_3"/>
( <font color="Maroon">m2'</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">m1'</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">m2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">m1</font> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K10">fComp</a> <span class="p2">(<span class="default"><a href="ens_1.html#K17">Hom</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m2'</font>,<font color="Maroon">m1'</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">m2'</font> <a href="ens_1.html#K6">*</a> <font color="Maroon">m1'</font>
<b>by </b><i>, <a class="ref" href="ens_1.html#D12">ENS_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , <a class="ref" href="ens_1.html#D7">ENS_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E10:2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">m1</font>
 ;<br/>


</div>
