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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/>



<b>assume </b><a NAME="E1:85"/><i><font color="Green">E62</font></i>: 
 <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:85"/>
<font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K1">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> #)
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D1">OPPCAT_1:def 1</a></i>;<br/>
<a NAME="E3:85"/><b>then </b>
(  <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K8" target="_self">fDom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9" target="_self">fCod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10" target="_self">fComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11" target="_self">fId</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:85"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E4:85"/><b>then </b>
( the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> &amp;  <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">c<sub>2</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="ens_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="ens_1.html#K23" target="_self">hom??</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E6:85"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span> holds  <a href="oppcat_1.html#K6">opp</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>;<br/>

<b>thus </b> <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K5">opp</a> 
<b>by </b><i><a class="ref" href="oppcat_1.html#D5">OPPCAT_1:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:85"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span> holds  <a href="oppcat_1.html#K4">opp</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>;<br/>

<b>thus </b> <a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K3">opp</a> 
<b>by </b><i><a class="ref" href="oppcat_1.html#D3">OPPCAT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="oppcat_1.html#D2">OPPCAT_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:85_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>, <font color="Maroon">c<sub>6</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:85_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K12">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E3:85_3_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ens_1.html#E1:85"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>8</sub></font> =  <a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_1_1"><b>now </b></a><div class="add"><b>thus </b> <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E2:85_3_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E6:85"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T22">OPPCAT_1:22</a></i>
;<br/><b>thus </b><a NAME="E2:85_3_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:85_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:85"><i><font color="Green">E62</font></i></a>, <a class="txt" href="ens_1.html#E92"><i><font color="Green">Lemma60</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:85_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">b<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"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">b<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"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>, <font color="Maroon">c<sub>6</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:85_3_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E3:85_3_2"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>, <font color="Maroon">c<sub>8</sub></font> =  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ens_1.html#E1:85"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E5:85_3_2"/>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E6:85"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E66</font></i>: <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="ens_1.html#E2:85_3_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="ens_1.html#D25" target="_self">Def25</a></i>
<br/>.= 
 <a href="ens_1.html#K16" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>
;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E2:85_3_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E6:85"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T22">OPPCAT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T12">OPPCAT_1:12</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E1:85"><i><font color="Green">E62</font></i></a>, <a class="txt" href="ens_1.html#E92"><i><font color="Green">Lemma60</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E6:85_3_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma6</font></i></a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E2:85_3_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E6:85"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T22">OPPCAT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T12">OPPCAT_1:12</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" target="_self">@</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E1:85"><i><font color="Green">E62</font></i></a>, <a class="txt" href="ens_1.html#E92"><i><font color="Green">Lemma60</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E6:85_3_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma6</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span>;<br/><b>assume </b><a NAME="E3:85_3"/><i><font color="Green">E65</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>, <font color="Maroon">c<sub>7</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:85_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>, <font color="Maroon">c<sub>9</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:85_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E8:85_3"/>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E6:85"><i><font color="Green">E63</font></i></a></i>;<br/><b>then </b><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="ens_1.html#E5:85_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="ens_1.html#D25" target="_self">Def25</a></i>
<br/>.= 
 <a href="ens_1.html#K16" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>
;<br/><a NAME="E10:85_3"/><b>then </b><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K22" target="_self">hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:85_3"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E12:85_3"/>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E6:85"><i><font color="Green">E63</font></i></a></i>;<br/><b>then </b><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ens_1.html#E7:85_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="ens_1.html#D25" target="_self">Def25</a></i>
<br/>.= 
 <a href="ens_1.html#K16" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>
;<br/><a NAME="E14:85_3"/><b>then </b><i><font color="Green">E70</font></i>: 
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K22" target="_self">hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E15:85_3"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="ens_1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E16:85_3"/><i><font color="Green">E72</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span> )
 <b>by </b><i><a class="txt" href="ens_1.html#E5:85_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="ens_1.html#E7:85_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/><a NAME="E17:85_3"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:85_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_5"><b>now </b></a><div class="add"><b>thus </b> <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E7:85"><i><font color="Green">E64</font></i></a>, <a class="txt" href="ens_1.html#E16:85_3"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T12">OPPCAT_1:12</a></i>
;<br/><b>thus </b> <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E7:85"><i><font color="Green">E64</font></i></a>, <a class="txt" href="ens_1.html#E16:85_3"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T12">OPPCAT_1:12</a></i>
;<br/></div><b>end;</b></div><a NAME="E19:85_3"/><b>then </b><i><font color="Green">E74</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E3:85_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E20:85_3"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E3:85_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="ens_1.html#E5:85_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="ens_1.html#E7:85_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="cat_2.html#T40">CAT_2:40</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E6:85"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E17:85_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="oppcat_1.html#T19">OPPCAT_1:19</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#D25" target="_self">Def25</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <span class="p4">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <span class="p4">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="ens_1.html#E19:85_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="ens_1.html#E20:85_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="ens_1.html#T56" target="_self">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E10:85_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="ens_1.html#E11:85_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="ens_1.html#E14:85_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="ens_1.html#E15:85_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="ens_1.html#E19:85_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="ens_1.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E10:85_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="ens_1.html#E14:85_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="ens_1.html#E19:85_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="ens_1.html#T28" target="_self">Th28</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:85"/>
 <a href="ens_1.html#K23" target="_self">hom??</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span>, <a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cat_1.html#T96">CAT_1:96</a></i>;<br/>


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