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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ringcat1.html#NM3">Ring_DOMAIN</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">c<sub>1</sub></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">c<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">c<sub>1</sub></font>] means  <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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:64"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="ringcat1.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:64"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ringcat1.html#E4:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:64"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E8:64"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="ringcat1.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E9:64"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E7:64"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ringcat1.html#E8:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E10:64"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="ringcat1.html#M2" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:64"><i><font color="Green">E42</font></i></a>, <a class="txt" href="ringcat1.html#E3:64"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ringcat1.html#E4:64"><i><font color="Green">E44</font></i></a>, <a class="txt" href="ringcat1.html#E5:64"><i><font color="Green">E45</font></i></a>, <a class="txt" href="ringcat1.html#E7:64"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ringcat1.html#E8:64"><i><font color="Green">E47</font></i></a></i>;<br/>


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