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

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

<b>set </b><font color="Maroon">C</font> =  <a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font>;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">UN</font>;<br/>
<b>set </b><font color="Maroon">X</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">UN</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="ringcat1.html#M2" target="_self">Element</a> of  <a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">UN</font> ;<br/>
<b>reconsider </b><font color="Maroon">b''</font> = <font color="Maroon">b'</font> as   <a href="funcsdom.html#NM1">Ring</a> ;<br/>
<a NAME="E3:76"/><i><font color="Green">E31</font></i>: 
 <a href="cat_1.html#K5">id</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K13" target="_self">ID</a> <font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><i><font color="Green">E32</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> = 
 <a href="ringcat1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">b''</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>

;<br/><br/>

<b>thus </b><i><font color="Green">E34</font></i>:  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> = 
 <a href="ringcat1.html#K2" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">b''</font></span>)</span>
<b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma52</font></i></a>, </i>
<br/>.= 
<font color="Maroon">b</font>

;<br/>

<b>thus </b><a NAME="E6:76"/>
for <font color="Olive">f</font> being  <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">UN</font></span>)</span>  st  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</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">UN</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:76_3"/><i><font color="Green">E35</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as  <a href="ringcat1.html#V2" target="_self">strict</a>  <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">UN</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f1</font> as  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a> ;<br/>
<a NAME="E4:76_3"/><i><font color="Green">E36</font></i>: 
 <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b''</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">b''</font></span>)</span> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f'</font>
<b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="ringcat1.html#T5" target="_self">Th5</a>, , <a class="txt" href="ringcat1.html#E15"><i><font color="Green">Lemma48</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="ringcat1.html#T6" target="_self">Th6</a>, <a class="ref" href="ringcat1.html#D2" target="_self">Def2</a></i>
;<br/>


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

<b>thus </b><a NAME="E7:76"/>
for <font color="Olive">g</font> being  <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">UN</font></span>)</span>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</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">UN</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:76_4"/><i><font color="Green">E37</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as  <a href="ringcat1.html#V2" target="_self">strict</a>  <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">UN</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f1</font> as  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a> ;<br/>
<a NAME="E4:76_4"/><i><font color="Green">E64</font></i>: 
 <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b''</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> = 
<font color="Maroon">f'</font> <a href="ringcat1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">b''</font></span>)</span>
<b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma52</font></i></a>, , , <a class="txt" href="ringcat1.html#E15"><i><font color="Green">Lemma48</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="ringcat1.html#T6" target="_self">Th6</a>, <a class="ref" href="ringcat1.html#D2" target="_self">Def2</a></i>
;<br/>


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


</div>
