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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="ringcat1.html#K10" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:73"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:73"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E3:73"/><i><font color="Green">E51</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:73"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E4:73"/><i><font color="Green">E52</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="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:73"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E5:73"/><i><font color="Green">E53</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:73"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self">Th29</a></i>;<br/>
<b>thus </b><a NAME="E6:73"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> iff  <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:73"><i><font color="Green">E49</font></i></a>, <a class="txt" href="ringcat1.html#E3:73"><i><font color="Green">E51</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self">Th29</a></i>;<br/>

<b>thus </b><a NAME="E7:73"/><i><font color="Green">E54</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K17" target="_self">comp</a> <span class="p2">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:73"><i><font color="Green">E50</font></i></a>, <a class="txt" href="ringcat1.html#E3:73"><i><font color="Green">E51</font></i></a>, <a class="ref" href="ringcat1.html#D23" target="_self">Def23</a></i>;<br/>

<b>thus </b><a NAME="E8:73"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green">E55</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:73_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ringcat1.html#T26" target="_self">Th26</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="ringcat1.html#K17" target="_self">comp</a> <span class="p2">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="ringcat1.html#E1:73"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="ringcat1.html#E7:73"><i><font color="Green">E54</font></i></a>, <a class="txt" href="ringcat1.html#E1:73_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ringcat1.html#D23" target="_self">Def23</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:73"/>
(  <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#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> iff  <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:73"><i><font color="Green">E49</font></i></a>, <a class="txt" href="ringcat1.html#E4:73"><i><font color="Green">E52</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self">Th29</a></i>;<br/>

<b>thus </b><a NAME="E10:73"/>
(  <a href="cat_1.html#K3">cod</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>6</sub></font> iff  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:73"><i><font color="Green">E49</font></i></a>, <a class="txt" href="ringcat1.html#E5:73"><i><font color="Green">E53</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self">Th29</a></i>;<br/>


</div>
