<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<a NAME="E1:32"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>6</sub></font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">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="ref" href="modcat_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:32"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:32"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:32"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>10</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is    <a href="grcat_1.html#M2">Element</a> of  <a href="grcat_1.html#K19">GroupObjects</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.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">c<sub>2</sub></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">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="modcat_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E6:32"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</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> &amp; ex <font color="Olive">b<sub>3</sub></font> being <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">b<sub>3</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">b<sub>3</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">b<sub>3</sub></font> #) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Olive">b<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="modcat_1.html#E5:32"><i><font color="Green">E24</font></i></a>, <a class="ref" href="modcat_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:32"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E8:32"/>
<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div>
