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

<b>let </b><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/>

<b>set </b><font color="Maroon" title="c2">C</font> =  <a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c3">V</font> =  <a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">f'</font>, <font color="Maroon" title="c8">g'</font> be    <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:74"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f'</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> )
 ;<br/>

<a NAME="E2:74"/><b>then </b><i><font color="Green" title="E50">A2</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c8">g'</font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#T29" target="_self" title="RINGCAT1:th.29">Th29</a></i>;<br/>
<a NAME="E3:74"/><i><font color="Green" title="E51">A3</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> <font color="Maroon" title="c7">f'</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:74"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self" title="RINGCAT1:th.29">Th29</a></i>;<br/>
<a NAME="E4:74"/><i><font color="Green" title="E52">A4</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c7">f'</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:74"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self" title="RINGCAT1:th.29">Th29</a></i>;<br/>
<a NAME="E5:74"/><i><font color="Green" title="E53">A5</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> <font color="Maroon" title="c8">g'</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E1:74"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self" title="RINGCAT1:th.29">Th29</a></i>;<br/>
<b>thus </b><a NAME="E6:74"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> iff  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> <font color="Maroon" title="c7">f'</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:74"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="ringcat1.html#E3:74"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self" title="RINGCAT1:th.29">Th29</a></i>;<br/>

<b>thus </b><a NAME="E7:74"/><i><font color="Green" title="E54">A6</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K17" title="RINGCAT1:func.17">comp</a> <span class="p2">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:74"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="ringcat1.html#E3:74"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="ringcat1.html#D23" target="_self" title="RINGCAT1:def.23">Def23</a></i>;<br/>

<b>thus </b><a NAME="E8:74"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c7">f'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green" title="E55">A7</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>
 ;<br/>

<a NAME="E2:74_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ringcat1.html#T26" target="_self" title="RINGCAT1:th.26">Th26</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="ringcat1.html#K17" title="RINGCAT1:func.17">comp</a> <span class="p2">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="ringcat1.html#E1:74"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">g'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="ringcat1.html#E7:74"><i><font color="Green" title="E54">A6</font></i></a>, <a class="txt" href="ringcat1.html#E1:74_1"><i><font color="Green" title="E55">A7</font></i></a>, <a class="ref" href="ringcat1.html#D23" target="_self" title="RINGCAT1:def.23">Def23</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E9:74"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> iff  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Maroon" title="c8">g'</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:74"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="ringcat1.html#E4:74"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self" title="RINGCAT1:th.29">Th29</a></i>;<br/>

<b>thus </b><a NAME="E10:74"/>
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g</font> iff  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> <font color="Maroon" title="c8">g'</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:74"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="ringcat1.html#E5:74"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="ringcat1.html#T29" target="_self" title="RINGCAT1:th.29">Th29</a></i>;<br/>


</div>
