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



<b>given </b><font color="Maroon">i</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><a NAME="E1:40"/><i><font color="Green">E38</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D11" target="_self">CAT_3:def 11</a><br/>

<b>assume </b><a NAME="E3:40"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="cat_1.html#V4">epi</a>
 ;<br/>

<b>take </b>
<font color="Maroon">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12">CAT_1:def 12</a><br/>

<b>thus </b><a NAME="E4:40"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>

<b>thus </b><i><font color="Green">E60</font></i>:  <a href="cat_1.html#K3">cod</a> <font color="Maroon">i</font> = 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:40_2"/>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>thus </b><a NAME="E2:40_2"/>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/><b>thus </b> <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i</font></span>)</span> = 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> = 
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:40"/>
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cat_1.html#D11">CAT_1:def 11</a></i>;<br/>

<b>thus </b><a NAME="E8:40"/>
<font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


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