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

<b>set </b><font color="Maroon" title="c3">C1</font> = <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c4">C2</font> = <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c1">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span> = <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default">$1 <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c1">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span><b> such that </b><br/><a NAME="E2:60_2_1"/><i><font color="Green" title="E38">A20</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> holds  <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:60_2_1"/><i><font color="Green" title="E39">A21</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:60_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:60_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c7">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:60_2_1_1"/><i><font color="Green" title="E40">A22</font></i>: 
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="txt" href="cat_5.html#E3:60_2_1_1"><i><font color="Green" title="E40">A22</font></i></a></i>;<br/>
<a NAME="E5:60_2_1_1"/><i><font color="Green" title="E41">A23</font></i>: 
<font color="Maroon" title="c6">x</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="cat_5.html#E3:60_2_1_1"><i><font color="Green" title="E40">A22</font></i></a></i>;<br/>
<a NAME="E6:60_2_1_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp; <font color="Maroon" title="c8">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T24" target="_self" title="CAT_5:th.24">Th24</a>, <a class="ref" href="cat_5.html#T31" target="_self" title="CAT_5:th.31">Th31</a></i>;<br/>
<a NAME="E7:60_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self" title="CAT_5:th.28">Th28</a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>;<br/>
<a NAME="E8:60_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_5.html#E5:60_2_1_1"><i><font color="Green" title="E41">A23</font></i></a>, <a class="ref" href="cat_5.html#D12" target="_self" title="CAT_5:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E9:60_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="txt" href="cat_5.html#E3:60_2_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<div><i><font color="Green" title="E40">A24</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" title="c7">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> st <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">d</font></span><br/><b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self" title="CAT_5:def.12">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c8">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self" title="CAT_5:th.28">Th28</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">d</font> = <font color="Maroon" title="c9">h</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self" title="CAT_5:def.12">Def12</a></i>;<br/><b>take </b><font color="Maroon" title="c11">d</font> = <font color="Maroon" title="c10">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">d</font></span><br/><a NAME="E4:60_2_1_2"/><i><font color="Green" title="E41">A25</font></i>: 
(  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">c</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="c7">c</font>,<font color="Maroon" title="c7">c</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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">d</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="c11">d</font>,<font color="Maroon" title="c11">d</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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">h</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="ref" href="cat_5.html#T32" target="_self" title="CAT_5:th.32">Th32</a></i>;<br/><a NAME="E5:60_2_1_2"/><b>then </b><i><font color="Green" title="E42">A26</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</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="c7">c</font></span>)</span> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</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="c7">c</font></span>)</span> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c8">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">d</font></span>)</span> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d</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="c11">d</font></span>)</span> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d</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="c11">d</font></span>)</span> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">h</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><a NAME="E6:60_2_1_2"/><i><font color="Green" title="E43">A27</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="cat_5.html#T24" target="_self" title="CAT_5:th.24">Th24</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">c</font></span>)</span> = 
<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="c9">h</font>,<font color="Maroon" title="c9">h</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> <font color="Maroon" title="c7">c</font></span>)</span> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="cat_5.html#E5:60_2_1_2"><i><font color="Green" title="E42">A26</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">d</font>
<b>by </b><i><a class="txt" href="cat_5.html#E4:60_2_1_2"><i><font color="Green" title="E41">A25</font></i></a>, <a class="txt" href="cat_5.html#E5:60_2_1_2"><i><font color="Green" title="E42">A26</font></i></a>, <a class="txt" href="cat_5.html#E6:60_2_1_2"><i><font color="Green" title="E43">A27</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E41">A28</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" title="c7">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> )</span><br/><b>reconsider </b><font color="Maroon" title="c8">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c9">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self" title="CAT_5:th.28">Th28</a></i>;<br/><a NAME="E2:60_2_1_3"/><i><font color="Green" title="E42">A29</font></i>: 
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="cat_5.html#T24" target="_self" title="CAT_5:th.24">Th24</a></i>;<br/><a NAME="E3:60_2_1_3"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</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="c8">g1</font>,<font color="Maroon" title="c9">g2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/><b>then </b> <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> = 
<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="c8">g1</font>,<font color="Maroon" title="c9">g2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self" title="CAT_5:th.31">Th31</a></i>
<br/>.= 
<font color="Maroon" title="c8">g1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/><a NAME="E5:60_2_1_3"/><b>then </b><i><font color="Green" title="E43">A30</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <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="c8">g1</font>,<font color="Maroon" title="c8">g1</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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c8">g1</font></span>)</span></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self" title="CAT_5:th.32">Th32</a></i>;<br/><a NAME="E6:60_2_1_3"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> 
 <b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self" title="CAT_5:th.31">Th31</a></i>;<br/><a NAME="E7:60_2_1_3"/><b>then </b>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">m</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span></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#K4" title="CAT_1:func.4">cod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self" title="CAT_5:th.32">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" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  &amp; <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#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  &amp; <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#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><b>hence </b><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> = 
<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="c8">g1</font>,<font color="Maroon" title="c8">g1</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#K4" title="CAT_1:func.4">cod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span></span>)</span></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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</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" title="E42">A29</font></i></a>, <a class="txt" href="cat_5.html#E5:60_2_1_3"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span></span><br/><a NAME="E10:60_2_1_3"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</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="c8">g1</font>,<font color="Maroon" title="c9">g2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/><b>then </b> <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span> = 
<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="c8">g1</font>,<font color="Maroon" title="c9">g2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self" title="CAT_5:th.31">Th31</a></i>
<br/>.= 
<font color="Maroon" title="c9">g2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/><a NAME="E12:60_2_1_3"/><b>then </b><i><font color="Green" title="E44">A31</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <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="c9">g2</font>,<font color="Maroon" title="c9">g2</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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">g2</font></span>)</span></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self" title="CAT_5:th.32">Th32</a></i>;<br/><a NAME="E13:60_2_1_3"/>
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> 
 <b>by </b><i><a class="ref" href="cat_5.html#T31" target="_self" title="CAT_5:th.31">Th31</a></i>;<br/><a NAME="E14:60_2_1_3"/><b>then </b>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></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#K4" title="CAT_1:func.4">cod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#T32" target="_self" title="CAT_5:th.32">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" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  &amp; <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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  &amp; <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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><b>hence </b><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> = 
<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="c9">g2</font>,<font color="Maroon" title="c9">g2</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#K4" title="CAT_1:func.4">cod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span></span>)</span></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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m</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" title="E42">A29</font></i></a>, <a class="txt" href="cat_5.html#E12:60_2_1_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c7">f1</font>, <font color="Maroon" title="c8">f2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f1</font> implies <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> )</span><br/><b>consider </b><font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c10">b1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> , <font color="Maroon" title="c11">g1</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="E2:60_2_1_4"/><i><font color="Green" title="E42">A32</font></i>: 
( <font color="Maroon" title="c7">f1</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="c9">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<font color="Maroon" title="c11">g1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c11">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self" title="CAT_5:def.12">Def12</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">a2</font>, <font color="Maroon" title="c13">b2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> , <font color="Maroon" title="c14">g2</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="E4:60_2_1_4"/><i><font color="Green" title="E43">A33</font></i>: 
( <font color="Maroon" title="c8">f2</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="c12">a2</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<font color="Maroon" title="c14">g2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c12">a2</font> &amp; <font color="Maroon" title="c14">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c12">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b2</font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D12" target="_self" title="CAT_5:def.12">Def12</a></i>;<br/><a NAME="E5:60_2_1_4"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  )
 <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="E6:60_2_1_4"/><b>then </b><i><font color="Green" title="E44">A34</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a2</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1_4"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="cat_5.html#E4:60_2_1_4"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">ha1</font> = <font color="Maroon" title="c9">a1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c16">ha2</font> = <font color="Maroon" title="c12">a2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c17">hb1</font> = <font color="Maroon" title="c10">b1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c18">hb2</font> = <font color="Maroon" title="c13">b2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a>  <b>by </b><i><a class="ref" href="cat_5.html#T28" target="_self" title="CAT_5:th.28">Th28</a></i>;<br/><a NAME="E8:60_2_1_4"/>
( <font color="Maroon" title="c7">f1</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c7">f1</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c7">f1</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Maroon" title="c8">f2</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a2</font> &amp; <font color="Maroon" title="c8">f2</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b2</font> &amp; <font color="Maroon" title="c8">f2</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g2</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1_4"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="cat_5.html#E4:60_2_1_4"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><a NAME="E9:60_2_1_4"/><b>then </b><i><font color="Green" title="E45">A35</font></i>: 
( <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f1</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="c15">ha1</font>,<font color="Maroon" title="c17">hb1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<font color="Maroon" title="c11">g1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f2</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="c16">ha2</font>,<font color="Maroon" title="c18">hb2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<font color="Maroon" title="c14">g2</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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/><b>assume </b><a NAME="E10:60_2_1_4"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f1</font></span>)</span></span><br/><a NAME="E11:60_2_1_4"/><b>then </b><i><font color="Green" title="E46">A36</font></i>: 
( <font color="Maroon" title="c8">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f1</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="c9">a1</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c11">g1</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> <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="c15">ha1</font>,<font color="Maroon" title="c18">hb2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c11">g1</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#E2:60_2_1_4"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="cat_5.html#E4:60_2_1_4"><i><font color="Green" title="E43">A33</font></i></a>, <a class="txt" href="cat_5.html#E6:60_2_1_4"><i><font color="Green" title="E44">A34</font></i></a>, <a class="txt" href="cat_5.html#E9:60_2_1_4"><i><font color="Green" title="E45">A35</font></i></a>, <a class="ref" href="cat_5.html#D12" target="_self" title="CAT_5:def.12">Def12</a></i>;<br/><a NAME="E12:60_2_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c11">g1</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><b>hence </b><a NAME="E13:60_2_1_4"/>
<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="cat_5.html#E11:60_2_1_4"><i><font color="Green" title="E46">A36</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="cat_5.html#E6:60_2_1"><i><font color="Green" title="E40">A24</font></i></a>, <a class="txt" href="cat_5.html#E7:60_2_1"><i><font color="Green" title="E41">A28</font></i></a>, <a class="ref" href="cat_1.html#T96" title="CAT_1:th.96">CAT_1:96</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> holds  <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span></span><br/>

<b>thus </b><a NAME="E10:60_2_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span> holds  <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </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#E2:60_2_1"><i><font color="Green" title="E38">A20</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
