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<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font>,   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font>] means  <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font>,   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="mod_2.html#L1">LModMorphismStr</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">a<sub>1</sub></font> <a href="mod_2.html#K8">*</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:28_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a>  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="modcat_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 <b>and </b><br/><a NAME="E4:28_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="grcat_1.html#S2">GRCAT_1:sch 2</a>(<a class="txt" href="modcat_1.html#E1:28_1_1"><i><font color="Green">E20</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E5:28_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> iff  <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">c<sub>2</sub></font> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="mod_2.html#K8">*</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="modcat_1.html#E3:28_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="modcat_1.html#E4:28_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>


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