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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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V7" target="_self">initial</a>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#R1" target="_self">are_isomorphic</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D16" target="_self">CAT_1:def 16</a><br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E4:71"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> holds <font color="Maroon">c<sub>5</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_1.html#E1:71"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#D16" target="_self">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:71"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:71"/>
<font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_1.html#E2:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#T81" target="_self">Th81</a></i>;<br/>
<a NAME="E8:71"/><i><font color="Green">E47</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E2:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#T81" target="_self">Th81</a></i>;<br/>
<a NAME="E9:71"/>
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:71"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#D16" target="_self">Def16</a></i>;<br/>
<b>hence </b><a NAME="E10:71"/><i><font color="Green">E48</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_1.html#E8:71"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cat_1.html#T51" target="_self">Th51</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E11:71"/><i><font color="Green">E49</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E2:71"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>

<b>thus </b><font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E6:71"><i><font color="Green">E46</font></i></a>, <a class="txt" href="cat_1.html#E10:71"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cat_1.html#T58" target="_self">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="cat_1.html#E8:71"><i><font color="Green">E47</font></i></a>, <a class="txt" href="cat_1.html#E10:71"><i><font color="Green">E48</font></i></a>, <a class="txt" href="cat_1.html#E11:71"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self">Th54</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="cat_1.html#E4:71"><i><font color="Green">E45</font></i></a></i>
;<br/>


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