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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">o</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_5.html#K14" title="CAT_5:func.14">-SliceCat</a> <font color="Maroon" title="c2">o</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c7">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E3:56"/><i><font color="Green" title="E37">A2</font></i>: 
(  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<font color="Maroon" title="c7">g</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D11" target="_self" title="CAT_5:def.11">Def11</a></i>;<br/>
<a NAME="E4:56"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">h</font> = <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c3">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="cat_5.html#K14" title="CAT_5:func.14">-SliceCat</a> <font color="Maroon" title="c2">o</font></span>)</span> <b>by </b><i><a class="ref" href="cat_5.html#D11" target="_self" title="CAT_5:def.11">Def11</a></i>;<br/>
<a NAME="E6:56"/>
( <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="cat_5.html#K17" title="CAT_5:func.17">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="cat_5.html#K18" title="CAT_5:func.18">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:56"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<a NAME="E7:56"/><b>then </b>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self" title="CAT_5:th.2">Th2</a></i>;<br/>
<a NAME="E8:56"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c8">h</font> = 
<font color="Maroon" title="c8">h</font> <a href="cat_5.html#K17" title="CAT_5:func.17">`11</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self" title="CAT_5:th.2">Th2</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="cat_5.html#E1:56"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c8">h</font> = 
<font color="Maroon" title="c8">h</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c3">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E1:56"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="cat_5.html#E3:56"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="cat_5.html#E8:56"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="cat_5.html#D11" target="_self" title="CAT_5:def.11">Def11</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c3">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<font color="Maroon" title="c7">g</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E1:56"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="cat_5.html#E3:56"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="cat_5.html#E8:56"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>
;<br/>
<b>hence </b><a NAME="E11:56"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:56"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="cat_5.html#E3:56"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="cat_5.html#E8:56"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/>


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