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<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>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:84"/><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> holds  <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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span>
 <b>and </b><br/><a NAME="E2:84"/><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">F</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="Olive">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="Olive">f</font></span>)</span> &amp; <font color="Maroon">F</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="Olive">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="Olive">f</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:84"/><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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:84_1"/>
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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>proof </b></a><div class="add">

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

<b>take </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
;<br/>

<b>thus </b><a NAME="E1:84_1_1"/>
<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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:84_1"/>
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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>thus </b><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="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" target="_self">dom</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#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>
<b>by </b><i/>
;<br/>

<b>thus </b><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> = 
 <a href="cat_1.html#K10" target="_self">id</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</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/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:84"/>
<font color="Maroon">T</font> is    <a href="cat_1.html#M2" target="_self">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>


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