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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> 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>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="cat_5.html#K13" target="_self">Hom</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:59"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="cat_5.html#K13" target="_self">Hom</a> , <font color="Maroon">c<sub>7</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:59"/><i><font color="Green">E39</font></i>: 
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K13">]</a></span>,<font color="Maroon">c<sub>7</sub></font></span><a href="cat_2.html#K13">]</a></span> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:59"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a>, <a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E6:59"/>
( <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_5.html#K20" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_5.html#K21" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:59"><i><font color="Green">E39</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/>
<a NAME="E7:59"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E8:59"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E1:59"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
;<br/>
<b>then </b><font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E1:59"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cat_5.html#E3:59"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cat_5.html#E8:59"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="cat_2.html#K13">]</a></span>,<font color="Maroon">c<sub>7</sub></font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E1:59"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cat_5.html#E3:59"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cat_5.html#E8:59"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>
;<br/>
<b>hence </b><a NAME="E11:59"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:59"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cat_5.html#E3:59"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cat_5.html#E8:59"><i><font color="Green">E40</font></i></a></i>;<br/>


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