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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM1" title="CAT_4:NM.1">Cartesian_category</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>



<a NAME="E1:41"/><i><font color="Green" title="E27">A1</font></i>: 
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T15" target="_self" title="CAT_4:th.15">Th15</a>, <a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/>
<a NAME="E2:41"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E3:41"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T14" target="_self" title="CAT_4:th.14">Th14</a>, <a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a>, <a class="ref" href="cat_1.html#T57" title="CAT_1:th.57">CAT_1:57</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
<span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <span class="p3">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <span class="p3">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="cat_4.html#E2:41"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="cat_4.html#T27" target="_self" title="CAT_4:th.27">Th27</a></i>;<br/>
<a NAME="E5:41"/>
<span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></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="c2">a</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="cat_4.html#D11" target="_self" title="CAT_4:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E6:41"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_4.html#K10" title="CAT_4:func.10">lambda</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/>

<a NAME="E7:41"/>
<span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E4:41"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="cat_4.html#T26" target="_self" title="CAT_4:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E8:41"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K10" title="CAT_4:func.10">lambda</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/>

<a NAME="E9:41"/><i><font color="Green" title="E30">A4</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<a NAME="E10:41"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T14" target="_self" title="CAT_4:th.14">Th14</a>, <a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a>, <a class="ref" href="cat_1.html#T57" title="CAT_1:th.57">CAT_1:57</a></i>;<br/>
<a NAME="E11:41"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
<span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="cat_4.html#E9:41"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="cat_4.html#T27" target="_self" title="CAT_4:th.27">Th27</a></i>;<br/>
<a NAME="E12:41"/>
<span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></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="c2">a</font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="cat_4.html#D11" target="_self" title="CAT_4:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E13:41"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_4.html#K12" title="CAT_4:func.12">rho</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/>

<a NAME="E14:41"/>
<span class="p1"><a href="cat_4.html#K9" title="CAT_4:func.9">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" title="CAT_4:func.6">term</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K9" title="CAT_4:func.9">:&gt;</a></span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</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"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E11:41"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="cat_4.html#T26" target="_self" title="CAT_4:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E15:41"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K12" title="CAT_4:func.12">rho</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/>


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