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<b>let </b><font color="Maroon">UN</font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>



<b>set </b><font color="Maroon">C</font> =  <a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="modcat_1.html#K4" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">f'</font> be    <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">b'</font> be    <a href="modcat_1.html#M2" target="_self">Element</a> of  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font>;<br/>







<a NAME="E1:32"/>
 ex <font color="Olive">G</font>, <font color="Olive">H</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="modcat_1.html#M2" target="_self">Element</a> of  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font> st <font color="Maroon">f</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Olive">G</font>,<font color="Olive">H</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:32"/>
<font color="Maroon">f</font> is  <a href="mod_2.html#V2">strict</a>  <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:32"/>
<font color="Maroon">f'</font> is   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:32"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">G</font>,<font color="Olive">ff</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">G</font> is    <a href="grcat_1.html#M2">Element</a> of  <a href="grcat_1.html#K19">GroupObjects</a> <font color="Maroon">UN</font>, <font color="Olive">ff</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> : verum </span>}</span>  &amp;  <a href="modcat_1.html#R1" target="_self">GO</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>,<font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:32"/>
 ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="tarski.html#K4">]</a></span> &amp;  ex <font color="Olive">G</font> being <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">G</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">G</font> #) &amp; <font color="Olive">x2</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Olive">G</font> ) )
 
<b>by </b><i><a class="ref" href="modcat_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E7:32"/>
<font color="Maroon">b</font> is  <a href="vectsp_1.html#V11">strict</a>  <a href="modcat_1.html#M2" target="_self">Element</a> of  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font>
 ;<br/>

<b>thus </b><a NAME="E8:32"/>
<font color="Maroon">b'</font> is   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>


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