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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_4.html#NM1">Cartesian_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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E25</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="E2:38"/><i><font color="Green">E26</font></i>: 
 <a href="cat_1.html#K6">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>and </b><br/><a NAME="E3:38"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="cat_1.html#V3">monic</a> or <font color="Maroon">c<sub>6</sub></font> is <a href="cat_1.html#V3">monic</a> )
 ;<br/>

<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="cat_1.html#V3">monic</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>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:38_1"/><i><font color="Green">E30</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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="E3:38_1"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E4:38_1"/>
( <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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_4.html#E1:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_4.html#E2:38"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cat_4.html#E2:38_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cat_4.html#T27" target="_self">Th27</a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E5:38_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:38_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_4.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E6:38_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#E1:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_4.html#E1:38_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_4.html#E2:38_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cat_1.html#T60">CAT_1:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:38"/><i><font color="Green">E29</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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#E1:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_4.html#E2:38"><i><font color="Green">E26</font></i></a>, <a class="ref" href="cat_4.html#T25" target="_self">Th25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="cat_1.html#V3">monic</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>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:38_2"/><i><font color="Green">E31</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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="E3:38_2"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E4:38_2"/>
( <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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_4.html#E1:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_4.html#E2:38"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cat_4.html#E2:38_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_4.html#T27" target="_self">Th27</a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E5:38_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:38_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="cat_4.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E6:38_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#E2:38"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cat_4.html#E1:38_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="cat_4.html#E2:38_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_1.html#T60">CAT_1:60</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:38"/>
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</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#K9" target="_self">:&gt;</a></span> is <a href="cat_1.html#V3">monic</a>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_4.html#E4:38"><i><font color="Green">E28</font></i></a>, <a class="txt" href="cat_4.html#E5:38"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cat_1.html#T60">CAT_1:60</a></i>;<br/>


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