<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="cat_2.html#K11">:]</a></span> = <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"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="cat_5.html#K19" target="_self">`2</a> </span>)</span></span><a href="cat_2.html#K13">]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">c<sub>1</sub></font></span><a href="cat_2.html#K11">:]</a></span><b> such that </b><br/><a NAME="E2:60_2_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:60_2_1"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:60_2_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_2_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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="txt" href="cat_5.html#E3:60_2_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E5:60_2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K19" 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#E2:60_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cat_5.html#E3:60_2_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E6:60_2_1_1"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K19" target="_self">`2</a> </span>)</span> <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>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K21" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K19" target="_self">`2</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T24" target="_self">Th24</a>, <a class="ref" href="cat_5.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E7:60_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</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> <a href="cat_5.html#K13" target="_self">Hom</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</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> <a href="cat_5.html#K13" target="_self">Hom</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K19" target="_self">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</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_5.html#K19" target="_self">`2</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self">Th28</a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>;<br/>
<a NAME="E8:60_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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="txt" href="cat_5.html#E5:60_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E9:60_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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="txt" href="cat_5.html#E3:60_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K13" target="_self">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K13" target="_self">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self">Th28</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E4:60_2_1_2"/><i><font color="Green">E41</font></i>: 
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</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>7</sub></font>,<font color="Maroon">c<sub>7</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>8</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</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>11</sub></font>,<font color="Maroon">c<sub>11</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>9</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E5:60_2_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</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>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</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>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_5.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</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>11</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</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>11</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="cat_5.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <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="E6:60_2_1_2"/><i><font color="Green">E43</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cat_5.html#T24" target="_self">Th24</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<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>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_5.html#K19" 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#E2:60_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cat_5.html#E5:60_2_1_2"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E4:60_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="cat_5.html#E5:60_2_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_5.html#E6:60_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K13" target="_self">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E2:60_2_1_3"/><i><font color="Green">E42</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="cat_5.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E3:60_2_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</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>8</sub></font>,<font color="Maroon">c<sub>9</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_5.html#K19" 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#E2:60_2_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>then </b> <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<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>8</sub></font>,<font color="Maroon">c<sub>9</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_5.html#K19" target="_self">`2</a> </span>)</span></span><a href="cat_2.html#K13">]</a></span> <a href="mcart_1.html#K17">`11</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self">Th31</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
;<br/><a NAME="E5:60_2_1_3"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <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>8</sub></font>,<font color="Maroon">c<sub>8</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>8</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E6:60_2_1_3"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> 
 <b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E7:60_2_1_3"/><b>then </b>
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E8:60_2_1_3"/><b>then </b>
( <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#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="cat_5.html#K20" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a>  &amp; <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#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="cat_5.html#K21" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a>  &amp; <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#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="cat_5.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span></span>)</span> )
 <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/><b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</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#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> = 
<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>8</sub></font>,<font color="Maroon">c<sub>8</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </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#E2:60_2_1"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_5.html#E5:60_2_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
;<br/><br/><a NAME="E10:60_2_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</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>8</sub></font>,<font color="Maroon">c<sub>9</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_5.html#K19" 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#E2:60_2_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>then </b> <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<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>8</sub></font>,<font color="Maroon">c<sub>9</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_5.html#K19" target="_self">`2</a> </span>)</span></span><a href="cat_2.html#K13">]</a></span> <a href="mcart_1.html#K18">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self">Th31</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
;<br/><a NAME="E12:60_2_1_3"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <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>9</sub></font>,<font color="Maroon">c<sub>9</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>9</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E13:60_2_1_3"/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> 
 <b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E14:60_2_1_3"/><b>then </b>
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E15:60_2_1_3"/><b>then </b>
( <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>7</sub></font></span>)</span></span>)</span> <a href="cat_5.html#K20" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a>  &amp; <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>7</sub></font></span>)</span></span>)</span> <a href="cat_5.html#K21" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a>  &amp; <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>7</sub></font></span>)</span></span>)</span> <a href="cat_5.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span></span>)</span> )
 <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/><b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</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>7</sub></font></span>)</span></span>)</span> = 
<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>9</sub></font>,<font color="Maroon">c<sub>9</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> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </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#E2:60_2_1"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_5.html#E12:60_2_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K13" target="_self">Hom</a> , <font color="Maroon">c<sub>11</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="E2:60_2_1_4"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>7</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>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_2.html#K13">]</a></span>,<font color="Maroon">c<sub>11</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>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K13" target="_self">Hom</a> , <font color="Maroon">c<sub>14</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="E4:60_2_1_4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>8</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>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="cat_2.html#K13">]</a></span>,<font color="Maroon">c<sub>14</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>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E5:60_2_1_4"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</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/><a NAME="E6:60_2_1_4"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_5.html#E4:60_2_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K13" target="_self">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E8:60_2_1_4"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K20" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K21" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="cat_5.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K20" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K21" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="cat_5.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_5.html#E4:60_2_1_4"><i><font color="Green">E43</font></i></a>, <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="E9:60_2_1_4"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</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>15</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="cat_2.html#K13">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="cat_2.html#K13">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</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>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="cat_2.html#K13">]</a></span>,<font color="Maroon">c<sub>14</sub></font></span><a href="cat_2.html#K13">]</a></span> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>assume </b><a NAME="E10:60_2_1_4"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E11:60_2_1_4"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</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>9</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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>15</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_5.html#E4:60_2_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="cat_5.html#E6:60_2_1_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="cat_5.html#E9:60_2_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cat_5.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E12:60_2_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</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>9</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</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>13</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_5.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>11</sub></font> )
 <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/><b>hence </b><a NAME="E13:60_2_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cat_5.html#E11:60_2_1_4"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E6:60_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cat_5.html#E7:60_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cat_1.html#T96">CAT_1:96</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E10:60_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_5.html#K15" target="_self">-SliceCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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"><span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cat_5.html#K20" target="_self">`11</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cat_5.html#K21" target="_self">`12</a> </span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cat_5.html#K19" 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#E2:60_2_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div>
