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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_4.html#NM1">Cartesian_category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:41"/><i><font color="Green">E27</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T15" target="_self">Th15</a>, <a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<a NAME="E2:41"/><i><font color="Green">E28</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E3:41"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K8" target="_self">pr2</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K7" target="_self">pr1</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T14" target="_self">Th14</a>, <a class="ref" href="cat_4.html#T21" target="_self">Th21</a>, <a class="ref" href="cat_1.html#T57">CAT_1:57</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <span class="p3">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <span class="p3">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_4.html#E2:41"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cat_4.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E5:41"/>
<span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="ref" href="cat_4.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b> <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_4.html#K10" target="_self">lambda</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cat_4.html#K10" target="_self">lambda</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K11" target="_self">lambda'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/><br/>

<a NAME="E7:41"/>
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E4:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cat_4.html#T26" target="_self">Th26</a></i>;<br/>
<b>hence </b> <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K10" target="_self">lambda</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cat_4.html#K11" target="_self">lambda'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K10" target="_self">lambda</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/><br/>

<a NAME="E9:41"/><i><font color="Green">E30</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_4.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E10:41"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T14" target="_self">Th14</a>, <a class="ref" href="cat_4.html#T21" target="_self">Th21</a>, <a class="ref" href="cat_1.html#T57">CAT_1:57</a></i>;<br/>
<a NAME="E11:41"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_4.html#E9:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cat_4.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E12:41"/>
<span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="ref" href="cat_4.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b> <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_4.html#K12" target="_self">rho</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cat_4.html#K12" target="_self">rho</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K13" target="_self">rho'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/><br/>

<a NAME="E14:41"/>
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E11:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_4.html#T26" target="_self">Th26</a></i>;<br/>
<b>hence </b> <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1"><a href="cat_4.html#K9" target="_self">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K6" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K9" target="_self">:&gt;</a></span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K12" target="_self">rho</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="cat_4.html#K13" target="_self">rho'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K12" target="_self">rho</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/><br/>


</div>
