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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_4.html#NM2">Cocartesian_category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <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>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:96"/><i><font color="Green">E56</font></i>: 
 <a href="cat_1.html#K6">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="E2:96"/><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</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="E3:96"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="cat_1.html#V4">epi</a> or <font color="Maroon">c<sub>6</sub></font> is <a href="cat_1.html#V4">epi</a> )
 ;<br/>

<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="cat_1.html#V4">epi</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</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>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:96_1"/><i><font color="Green">E61</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</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="E3:96_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span>
 ;<br/><a NAME="E4:96_1"/>
( <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span> &amp; <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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_4.html#E1:96"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_4.html#E2:96"><i><font color="Green">E57</font></i></a>, <a class="txt" href="cat_4.html#E2:96_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self">Th72</a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E5:96_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:96_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="cat_4.html#T73" target="_self">Th73</a></i>;<br/><b>hence </b><a NAME="E6:96_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:96"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_4.html#E1:96_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="cat_4.html#E2:96_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="cat_1.html#T65">CAT_1:65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:96"/><i><font color="Green">E60</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<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_4.html#E1:96"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_4.html#E2:96"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_4.html#T70" target="_self">Th70</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="cat_1.html#V4">epi</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</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>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:96_2"/><i><font color="Green">E62</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</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="E3:96_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span>
 ;<br/><a NAME="E4:96_2"/>
( <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span> &amp; <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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_4.html#E1:96"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_4.html#E2:96"><i><font color="Green">E57</font></i></a>, <a class="txt" href="cat_4.html#E2:96_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self">Th72</a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E5:96_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:96_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="cat_4.html#T73" target="_self">Th73</a></i>;<br/><b>hence </b><a NAME="E6:96_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E2:96"><i><font color="Green">E57</font></i></a>, <a class="txt" href="cat_4.html#E1:96_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="cat_4.html#E2:96_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="cat_1.html#T65">CAT_1:65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:96"/>
<span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_4.html#K27" target="_self">$]</a></span> is <a href="cat_1.html#V4">epi</a>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:96"><i><font color="Green">E58</font></i></a>, <a class="txt" href="cat_4.html#E4:96"><i><font color="Green">E59</font></i></a>, <a class="txt" href="cat_4.html#E5:96"><i><font color="Green">E60</font></i></a>, <a class="ref" href="cat_1.html#T65">CAT_1:65</a></i>;<br/>


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