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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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:61"/><i><font color="Green">E40</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:61"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="cat_5.html#K14" target="_self">-SliceCat</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="cat_5.html#K14" target="_self">-SliceCat</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font> <a href="cat_5.html#K14" target="_self">-SliceCat</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="cat_5.html#K14" target="_self">-SliceCat</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="cat_5.html#E1:61"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="cat_5.html#K22" target="_self">SliceFunctor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font> <a href="cat_5.html#K14" target="_self">-SliceCat</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="cat_5.html#K14" target="_self">-SliceCat</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="cat_5.html#E2:61"><i><font color="Green">E41</font></i></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>9</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cat_5.html#K14" target="_self">-SliceCat</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/><a NAME="E1:61_1"/>
<span class="p1">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K17" target="_self">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K18" target="_self">`12</a> </span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K16" target="_self">`2</a> </span>)</span></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E2:61_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K17" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K17" target="_self">`11</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K18" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K18" target="_self">`12</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K16" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K16" target="_self">`2</a>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/><a NAME="E3:61_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K17" target="_self">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K18" target="_self">`12</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="cat_5.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E4:61_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K17" target="_self">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K17" target="_self">`11</a> </span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K18" target="_self">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K18" target="_self">`12</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:61"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K14">*</a> <span class="p2">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K17" target="_self">`11</a> </span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K18" target="_self">`12</a> </span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K16" target="_self">`2</a> </span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E1:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cat_5.html#E2:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_5.html#D13" target="_self">Def13</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="cat_5.html#E4:61_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_5.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:61"/>
 <a href="cat_5.html#K22" target="_self">SliceFunctor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="cat_5.html#K22" target="_self">SliceFunctor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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