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<span class="kw">let </span><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/>  for <font color="Olive" title="b2">b</font> being   <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="Olive" title="b1">R</font>)</span>)</span> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; (  for <font color="Olive" title="b3">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="Olive" title="b1">R</font>)</span>)</span>  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) &amp; (  for <font color="Olive" title="b3">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="Olive" title="b1">R</font>)</span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <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> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; (  for <font color="Olive" title="b2">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="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <font color="Olive" title="b2">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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/>



<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; (  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="graph_1.html#K2" title="GRAPH_1:func.2">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#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> ) &amp; (  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="graph_1.html#K1" title="GRAPH_1:func.1">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#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">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> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">b9</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">b99</font> = <font color="Maroon" title="c7">b9</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="E3:39"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">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">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#T14" target="_self" title="MODCAT_1:th.14">Th16</a></span>;<br/>
<span class="kw">hence </span><span class="lab"><font color="Green" title="E31">A2</font></span>:  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">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">b99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#T15" target="_self" title="MODCAT_1:th.15">Th17</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#T18" title="MOD_2:th.18">MOD_2:18</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; (  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="graph_1.html#K2" title="GRAPH_1:func.2">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#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> ) &amp; (  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="graph_1.html#K1" title="GRAPH_1:func.1">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#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">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> ) )</span><br/>

<span class="kw">thus </span><span class="lab"><font color="Green" title="E32">A3</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">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">b99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="modcat_1.html#T15" target="_self" title="MODCAT_1:th.15">Th17</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#T18" title="MOD_2:th.18">MOD_2:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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="graph_1.html#K2" title="GRAPH_1:func.2">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#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> ) &amp; (  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="graph_1.html#K1" title="GRAPH_1:func.1">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#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">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> ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:39"/>
 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="graph_1.html#K2" title="GRAPH_1:func.2">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#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="graph_1.html#K1" title="GRAPH_1:func.1">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#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">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></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">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> implies <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_3"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font></span><br/>

<span class="kw">reconsider </span><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="E3:39_3"/>
 ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D7" target="_self" title="MODCAT_1:def.7">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f9</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="E5:39_3"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c11">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_3"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="modcat_1.html#T15" target="_self" title="MODCAT_1:th.15">Th17</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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">b99</font></span>)</span> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c11">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E1:39_3"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="modcat_1.html#T16" target="_self" title="MODCAT_1:th.16">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_3"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="mod_2.html#T18" title="MOD_2:th.18">MOD_2:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:39"/>
 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="graph_1.html#K1" title="GRAPH_1:func.1">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#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">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> implies <font color="Maroon" title="c9">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_4"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font></span><br/>

<span class="kw">reconsider </span><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="E3:39_4"/>
 ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D7" target="_self" title="MODCAT_1:def.7">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f9</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="E5:39_4"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c11">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_4"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="modcat_1.html#T15" target="_self" title="MODCAT_1:th.15">Th17</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c9">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c6">b</font></span>)</span> = 
<font color="Maroon" title="c11">f9</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">b99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E1:39_4"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="modcat_1.html#T16" target="_self" title="MODCAT_1:th.16">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_4"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="mod_2.html#T18" title="MOD_2:th.18">MOD_2:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
