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

<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>let </b><font color="Maroon">c<sub>4</sub></font> be    <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>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green">E37</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>and </b><br/><a NAME="E2:61"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="cat_1.html#V5" target="_self">invertible</a>
 ;<br/>

<a NAME="E3:61"/><i><font color="Green">E39</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#E1:61"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cat_1.html#E2:61"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self">Th70</a></i>;<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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:61"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>5</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="E6:61"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>4</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#E1:61"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cat_1.html#E2:61"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self">Th70</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</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>7</sub></font>, <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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_1"/><i><font color="Green">E42</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c<sub>6</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>and </b><br/><a NAME="E2:61_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</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> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>4</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#E6:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="cat_1.html#E1:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T57" target="_self">Th57</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:61"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cat_1.html#E3:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cat_1.html#E1:61_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_1.html#E2:61_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="cat_1.html#E1:61"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cat_1.html#E3:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cat_1.html#E6:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="cat_1.html#E1:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self">Th54</a></i>
;<br/><b>hence </b><a NAME="E4:61_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:61"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="cat_1.html#V3" target="_self">monic</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:61"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cat_1.html#T60" target="_self">Th60</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</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>7</sub></font>, <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>6</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_2"/><i><font color="Green">E42</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>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:61_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E5:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cat_1.html#E1:61_2"><i><font color="Green">E42</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>4</sub></font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="cat_1.html#E1:61"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cat_1.html#E3:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cat_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_1.html#E2:61_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self">Th54</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:61"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cat_1.html#E3:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cat_1.html#E5:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cat_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self">Th54</a></i>
;<br/><b>hence </b><a NAME="E4:61_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T58" target="_self">Th58</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:61"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="cat_1.html#V4" target="_self">epi</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:61"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cat_1.html#T65" target="_self">Th65</a></i>;<br/>


</div>
