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<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>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>, <font color="Maroon">g</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>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <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> ;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <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> ;<br/>
<i><font color="Green">E31</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="ringcat1.html#K14" target="_self">dom</a> <span class="p2">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">UN</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">g'</font>
<b>by </b><i/>
;<br/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="ringcat1.html#K15" target="_self">cod</a> <span class="p2">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">UN</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">f'</font>
<b>by </b><i><a class="txt" href="ringcat1.html#E9"><i><font color="Green">Lemma39</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:69"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</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">UN</font></span>)</span> iff  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></i>;<br/>


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