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

<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">T</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  ex <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> st <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Olive">d</font>
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds <br/> ( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E3:81"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">f</font> holds <br/><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E4:81"/>
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>  ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font> st <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D18" target="_self">CAT_1:def 18</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">d</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:81_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:81_1"/>
 ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font> st <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:81"/>
for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font> holds <br/> ( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:81_2"/>
( the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">f</font> &amp; the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">f</font></span>)</span> &amp; the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> &amp; the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> &amp; the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font></span>)</span> &amp; the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> &amp; the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E2:81_2"/>
( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E6:81"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font>
 ;<br/>

<a NAME="E7:81"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>then </b> <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E9:81"/><b>then </b>
 <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_1.html#E3:81"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E10:81"/><b>then </b>
( the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> &amp; the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#D8" target="_self">Def8</a>, </i>;<br/>
<b>hence </b><a NAME="E11:81"/>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="cat_1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
