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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:102"/><i><font color="Green">E58</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:102"/><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:102"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) iff <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) ) )
 <b>and </b><br/><a NAME="E4:102"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:102"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D19" target="_self">CAT_3:def 19</a><br/>

<a NAME="E6:102"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:102"><i><font color="Green">E58</font></i></a>, <a class="txt" href="cat_3.html#E2:102"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:102"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:102"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) iff <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:102"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E10:102"/>
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:102"><i><font color="Green">E58</font></i></a>, <a class="txt" href="cat_3.html#E2:102"><i><font color="Green">E59</font></i></a>, <a class="ref" href="cat_1.html#T46">CAT_1:46</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<sub>2</sub></font> = 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E9:102"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E4:102"><i><font color="Green">E61</font></i></a>, <a class="txt" href="cat_3.html#E5:102"><i><font color="Green">E62</font></i></a>, <a class="txt" href="cat_3.html#E9:102"><i><font color="Green">E63</font></i></a></i>
;<br/><br/>


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