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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="ringcat1.html#NM3" title="RINGCAT1:NM.3">Ring_DOMAIN</a>;<br/>

<b>set </b><font color="Maroon" title="c2">X</font> =  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font>,   <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font>] <b>means </b> <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> $2;<br/>
<b>let </b><font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">f</font> be    <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:66"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">f</font>]
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font>, <font color="Maroon" title="c7">G3</font> being    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:66"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c5">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c6">G2</font> &amp; <font color="Maroon" title="c6">G2</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c7">G3</font> &amp; <font color="Maroon" title="c3">g</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c7">G3</font> &amp; <font color="Maroon" title="c4">f</font> is    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font> )
 <b>by </b><i><a class="ref" href="ringcat1.html#T24" target="_self" title="RINGCAT1:th.24">Th24</a></i>;<br/>
<a NAME="E4:66"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c5">G1</font> <a href="ringcat1.html#R1" title="RINGCAT1:pred.1">&lt;=</a> <font color="Maroon" title="c7">G3</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:66"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="ringcat1.html#T8" target="_self" title="RINGCAT1:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g'</font> = <font color="Maroon" title="c3">g</font> as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c7">G3</font> <b>by </b><i><a class="txt" href="ringcat1.html#E3:66"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">f'</font> = <font color="Maroon" title="c4">f</font> as    <a href="ringcat1.html#M1" title="RINGCAT1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font> <b>by </b><i><a class="txt" href="ringcat1.html#E3:66"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>
<a NAME="E7:66"/>
<font color="Maroon" title="c8">g'</font> <a href="ringcat1.html#K7" title="RINGCAT1:func.7">*'</a> <font color="Maroon" title="c9">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c9">f'</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:66"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="ringcat1.html#D11" target="_self" title="RINGCAT1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E8:66"/>
<font color="Maroon" title="c3">g</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E4:66"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="ringcat1.html#D18" target="_self" title="RINGCAT1:def.18">Def18</a></i>;<br/>


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