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

<b>let </b><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/>



<b>set </b><font color="Maroon" title="c3">C</font> =  <a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">V</font> =  <a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> =  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c6">b</font> as    <a href="modcat_1.html#M2" title="MODCAT_1:mode.2">Element</a> of  <a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">b''</font> = <font color="Maroon" title="c7">b'</font> as   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">R</font> ;<br/>
<a NAME="E3:38"/><i><font color="Green" title="E30">A1</font></i>: 
 <a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K7" title="MODCAT_1:func.7">ID</a> <font color="Maroon" title="c7">b'</font>
 
<b>by </b><i><a class="ref" href="modcat_1.html#T16" target="_self" title="MODCAT_1:th.16">Th16</a></i>;<br/>
<b>hence </b><i><font color="Green" title="E31">A2</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c6">b</font></span>)</span> = 
 <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <span class="p1">(<span class="default"><a href="mod_2.html#K6" title="MOD_2:func.6">ID</a> <font color="Maroon" title="c8">b''</font></span>)</span>
<b>by </b><i><a class="ref" href="modcat_1.html#T17" target="_self" title="MODCAT_1:th.17">Th17</a></i>
<br/>.= 
<font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="mod_2.html#T26" title="MOD_2:th.26">MOD_2:26</a></i>
;<br/><br/>

<b>thus </b><i><font color="Green" title="E32">A3</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c6">b</font></span>)</span> = 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <span class="p1">(<span class="default"><a href="mod_2.html#K6" title="MOD_2:func.6">ID</a> <font color="Maroon" title="c8">b''</font></span>)</span>
<b>by </b><i><a class="txt" href="modcat_1.html#E3:38"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self" title="MODCAT_1:th.17">Th17</a></i>
<br/>.= 
<font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="mod_2.html#T26" title="MOD_2:th.26">MOD_2:26</a></i>
;<br/>

<b>thus </b><a NAME="E6:38"/>
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:38_3"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">f1</font> = <font color="Maroon" title="c9">f</font> as    <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span> ;<br/>
<a NAME="E3:38_3"/>
 ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="modcat_1.html#M2" title="MODCAT_1:mode.2">Element</a> of  <a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font> st <font color="Maroon" title="c10">f1</font> is  <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>  <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font>
 
<b>by </b><i><a class="ref" href="modcat_1.html#D7" target="_self" title="MODCAT_1:def.7">Def7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">f'</font> = <font color="Maroon" title="c10">f1</font> as  <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a> <a href="mod_2.html#NM1" title="MOD_2:NM.1">LModMorphism</a> of <font color="Maroon" title="c2">R</font> ;<br/>
<a NAME="E5:38_3"/><i><font color="Green" title="E34">A5</font></i>: 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b''</font>
 
<b>by </b><i><a class="txt" href="modcat_1.html#E1:38_3"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self" title="MODCAT_1:th.17">Th17</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">f</font> = 
<span class="p1">(<span class="default"><a href="mod_2.html#K6" title="MOD_2:func.6">ID</a> <font color="Maroon" title="c8">b''</font></span>)</span> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c11">f'</font>
<b>by </b><i><a class="txt" href="modcat_1.html#E3:38"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="modcat_1.html#E4:38"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="modcat_1.html#E1:38_3"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="modcat_1.html#T18" target="_self" title="MODCAT_1:th.18">Th18</a></i>
<br/>.= 
<font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="modcat_1.html#E5:38_3"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="mod_2.html#T26" title="MOD_2:th.26">MOD_2:26</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:38"/>
for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" title="MODCAT_1:func.12">LModCat</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:38_4"/><i><font color="Green" title="E33">A6</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">f1</font> = <font color="Maroon" title="c9">f</font> as    <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font></span>)</span> ;<br/>
<a NAME="E3:38_4"/>
 ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="modcat_1.html#M2" title="MODCAT_1:mode.2">Element</a> of  <a href="modcat_1.html#K3" title="MODCAT_1:func.3">LModObjects</a> <font color="Maroon" title="c1">UN</font>,<font color="Maroon" title="c2">R</font> st <font color="Maroon" title="c10">f1</font> is  <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>  <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font>
 
<b>by </b><i><a class="ref" href="modcat_1.html#D7" target="_self" title="MODCAT_1:def.7">Def7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">f'</font> = <font color="Maroon" title="c10">f1</font> as  <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a> <a href="mod_2.html#NM1" title="MOD_2:NM.1">LModMorphism</a> of <font color="Maroon" title="c2">R</font> ;<br/>
<a NAME="E5:38_4"/><i><font color="Green" title="E34">A7</font></i>: 
 <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b''</font>
 
<b>by </b><i><a class="txt" href="modcat_1.html#E1:38_4"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self" title="MODCAT_1:th.17">Th17</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c9">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c6">b</font></span>)</span> = 
<font color="Maroon" title="c11">f'</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <span class="p1">(<span class="default"><a href="mod_2.html#K6" title="MOD_2:func.6">ID</a> <font color="Maroon" title="c8">b''</font></span>)</span>
<b>by </b><i><a class="txt" href="modcat_1.html#E3:38"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="modcat_1.html#E5:38"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="modcat_1.html#E1:38_4"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="modcat_1.html#T18" target="_self" title="MODCAT_1:th.18">Th18</a></i>
<br/>.= 
<font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="modcat_1.html#E5:38_4"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="mod_2.html#T26" title="MOD_2:th.26">MOD_2:26</a></i>
;<br/>


</div><b>end;</b></div>


</div>
