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<b>let </b><font color="Maroon">V</font> be   <a href="ringcat1.html#NM3">Ring_DOMAIN</a>;<br/>

<b>set </b><font color="Maroon">X</font> =  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">V</font>,   <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">V</font>] <b>means </b> <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">g</font> be    <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">V</font>, <font color="Maroon">f</font> be    <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:65"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>,<font color="Maroon">f</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">G1</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">G2</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">G3</font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:65"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">G1</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">G2</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G3</font> &amp; <font color="Maroon">g</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font> &amp; <font color="Maroon">f</font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:65"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">G1</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G3</font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self">Def8</a>, </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 <font color="Maroon">G2</font>,<font color="Maroon">G3</font> <b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></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">G1</font>,<font color="Maroon">G2</font> <b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E7:65"/>
<font color="Maroon">g'</font> <a href="ringcat1.html#K7" target="_self">*'</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f'</font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self">Def8</a>, </i>;<br/>
<b>hence </b><a NAME="E8:65"/>
<font color="Maroon">g</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E13"><i><font color="Green">Lemma44</font></i></a>, </i>;<br/>


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