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

<b>let </b><font color="Maroon">G</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>

<b>set </b><font color="Maroon">i</font> =  <a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">G</font>;<br/>
<b>thus </b><a NAME="E1:34"/><i><font color="Green">E31</font></i>: 
(  <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">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp;  <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">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 ;<br/>

<b>thus </b><a NAME="E2:34"/>
for <font color="Olive">f</font> being <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>  st  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="ringcat1.html#K6" target="_self">*</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="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E32</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">G</font>
 ;<br/>

<b>set </b><font color="Maroon">H</font> =  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font>;<br/>
<a NAME="E2:34_1"/><i><font color="Green">E34</font></i>: 
 <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font>,<font color="Maroon">G</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:34_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <span class="p1">(<span class="default"><a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">G</font>,<font color="Maroon">m</font> #)
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:34_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">G</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">m</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1"/>
<span class="p1">(<span class="default"><a href="ringcat1.html#K5" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="ringcat1.html#T3" target="_self">Th3</a>, , , </i>;<br/>


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

<b>thus </b><a NAME="E3:34"/>
for <font color="Olive">g</font> being <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>  st  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> holds <br/><font color="Olive">g</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">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E36</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">G</font>
 ;<br/>

<b>set </b><font color="Maroon">H</font> =  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>;<br/>
<a NAME="E2:34_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">G</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>, <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E5:34_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">m</font> #)
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:34_2"/>
<font color="Maroon">m</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E7:34_2"/>
<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">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="ringcat1.html#T3" target="_self">Th3</a>, , , </i>;<br/>


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


</div>
