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<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>deffunc </b><font color="Maroon">H<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>-&gt; </b> <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a> <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> = $1 <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> $2;<br/>
<a NAME="E1:66_1_1"/><i><font color="Green" title="E43">A1</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">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="ref" href="ringcat1.html#T25" target="_self" title="RINGCAT1:th.25">Th25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">c</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:66_1_1"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <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> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">c</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font>] )
 <b>and </b><br/><a NAME="E4:66_1_1"/><i><font color="Green" title="E45">A3</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <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>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">c</font> holds <br/><font color="Maroon" title="c3">c</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S8" title="BINOP_1:sch.8">BINOP_1:sch 8</a>(<a class="txt" href="ringcat1.html#E1:66_1_1"><i><font color="Green" title="E43">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">c</font>
;<br/>

<b>thus </b><a NAME="E5:66_1_1"/>
( ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <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> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">c</font> iff  <a href="ringcat1.html#K11" title="RINGCAT1:func.11">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K12" title="RINGCAT1:func.12">cod</a> <font color="Olive" title="b2">f</font> ) ) &amp; ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <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>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">c</font> holds <br/><font color="Maroon" title="c3">c</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Olive" title="b2">f</font> ) )
 <b>by </b><i><a class="txt" href="ringcat1.html#E3:66_1_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="ringcat1.html#E4:66_1_1"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/>


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