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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/>





<b>given </b><font color="Maroon" title="c4">i</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><a NAME="E1:36"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D11" target="_self" title="CAT_3:def.11">CAT_3:def 11</a><br/>

<b>given </b><font color="Maroon" title="c5">j</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><a NAME="E3:36"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c3">g</font>
 <b>and </b><br/><a NAME="E4:36"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D11" target="_self" title="CAT_3:def.11">CAT_3:def 11</a><br/>

<b>assume </b><a NAME="E5:36"/><i><font color="Green" title="E28">A5</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<b>take </b>
<font color="Maroon" title="c4">i</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">j</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D11" target="_self" title="CAT_3:def.11">CAT_3:def 11</a><br/>

<i><font color="Green" title="E29">A6</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">j</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c3">g</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E3:36"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">g</font>
<b>by </b><i><a class="txt" href="cat_3.html#E4:36"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
;<br/>
<a NAME="E7:36"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_3.html#E5:36"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/>
<b>hence </b><a NAME="E8:36"/>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:36"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="cat_3.html#E3:36"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="cat_3.html#E5:36"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="cat_3.html#E6:36"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
<font color="Maroon" title="c4">i</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:36"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="cat_3.html#E3:36"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="cat_3.html#E5:36"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="cat_3.html#E6:36"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="cat_3.html#E7:36"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c4">i</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E3:36"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="cat_3.html#E5:36"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E2:36"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="cat_3.html#E4:36"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="cat_3.html#E5:36"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E5:36"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
;<br/>


</div>
