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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">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E38</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E42</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E3:74"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> 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 <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span> holds <br/> ex <font color="Olive">h</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<font color="Maroon">c</font> &amp; ( for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<font color="Maroon">c</font> holds <br/>( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) iff <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> ) ) )
 <b>and </b><br/><a NAME="E4:74"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E5:74"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self">CAT_3:def 16</a><br/>

<a NAME="E6:74"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E3:74"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </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><br/><a NAME="E8:74"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E9:74"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> holds <br/>( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> ) iff <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:74"/>
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E3:74"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_3.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T47">CAT_1:47</a>, <a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<b>hence </b> <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font> = 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="cat_3.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i>, , <a class="ref" href="cat_3.html#T2" target="_self">Th2</a></i>
;<br/><br/>


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