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<b>set </b><font color="Maroon" title="c2">X</font> =  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>,   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>] <b>means </b> <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> $2;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>,   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>) <b>-&gt; </b> <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a> <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a> = $1 <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> $2;<br/>
<a NAME="E1:87_1_1"/><i><font color="Green" title="E48">A1</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T45" target="_self" title="GRCAT_1:th.45">Th45</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="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:87_1_1"/><i><font color="Green" title="E49">A2</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="relat_1.html#NK2" title="RELAT_1:NK.2">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="E4:87_1_1"/><i><font color="Green" title="E50">A3</font></i>: 
for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E1:87_1_1"><i><font color="Green" title="E48">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c</font> iff  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">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="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="relat_1.html#NK2" title="RELAT_1:NK.2">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="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<b>thus </b><a NAME="E5:87_1_1"/>
( ( for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c</font> iff  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">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="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">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="relat_1.html#NK2" title="RELAT_1:NK.2">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="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Olive" title="b2">f</font> ) )
 <b>by </b><i><a class="txt" href="#E3:87_1_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E4:87_1_1"><i><font color="Green" title="E50">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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