<?xml version="1.0"?>
<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>, <font color="Maroon">g</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>assume </b><a NAME="E1:30"/><i><font color="Green">E31</font></i>: 
 <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>
 ;<br/>

<b>set </b><font color="Maroon">G1</font> =  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">G2</font> =  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">G3</font> =  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font>;<br/>
<a NAME="E2:30"/><i><font color="Green">E32</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>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a>  <a href="ringcat1.html#K2" target="_self">cod</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>, <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>, <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">f0</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>,<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="E6:30"/><i><font color="Green">E34</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>,<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">f0</font> #)
 ;<br/>
<b>consider </b><font color="Maroon">g0</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font></span>)</span><b> such that </b><br/><a NAME="E8:30"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g'</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#K2" target="_self">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g0</font> #)
 <b>by </b><i>, </i>;<br/>
<b>take </b>
 <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font>
;<br/>

<b>take </b>
 <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>
;<br/>

<b>take </b>
 <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font>
;<br/>

<b>take </b>
<font color="Maroon">f0</font>
;<br/>

<b>take </b>
<font color="Maroon">g0</font>
;<br/>

<b>thus </b><a NAME="E9:30"/>
( <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>,<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">f0</font> #) &amp; <font color="Maroon">g</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#K2" target="_self">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">g0</font> #) &amp; <font color="Maroon">g</font> <a href="ringcat1.html#K6" target="_self">*</a> <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>,<span class="p1">(<span class="default"><a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f0</font></span>)</span> #) )
 <b>by </b><i>, , <a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, </i>;<br/>


</div>
