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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>, <font color="Maroon">c<sub>5</sub></font>, <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> 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>8</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>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E32</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="E2:57"/><i><font color="Green">E33</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="E3:57"/><i><font color="Green">E34</font></i>: 
 <a href="cat_1.html#K6">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="E4:57"/><i><font color="Green">E35</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>6</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> 
 ;<br/>

<a NAME="E5:57"/><i><font color="Green">E36</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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#E3:57"><i><font color="Green">E34</font></i></a>, <a class="txt" href="cat_4.html#E4:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="cat_4.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E6:57"/>
(  <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#K2" target="_self">[x]</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<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>  &amp;  <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#K2" target="_self">[x]</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<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>  &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E3:57"><i><font color="Green">E34</font></i></a>, <a class="txt" href="cat_4.html#E4:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="cat_4.html#D11" target="_self">Def11</a>, <a class="ref" href="cat_4.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E7:57"/><b>then </b>
(  <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#K2" target="_self">[x]</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>  &amp;  <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#K2" target="_self">[x]</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:57"><i><font color="Green">E32</font></i></a>, <a class="txt" href="cat_4.html#E2:57"><i><font color="Green">E33</font></i></a>, <a class="txt" href="cat_4.html#E5:57"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a>, <a class="ref" href="cat_1.html#T54">CAT_1:54</a></i>;<br/>
<a NAME="E8:57"/><b>then </b>
<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>7</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>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>10</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E5:57"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cat_4.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E9:57"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_4.html#K18" target="_self">[x]</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</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>8</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span>
 ;<br/>


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