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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">f</font>, <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> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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> iff  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b2">f</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <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> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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> iff  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font> )</span><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>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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> iff  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c5">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/>
<b>reconsider </b><font color="Maroon" title="c8">g'</font> = <font color="Maroon" title="c6">g</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/>
<i><font color="Green" title="E23">A1</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> = 
<span class="p1">(<span class="default"><a href="modcat_1.html#K8" title="MODCAT_1:func.8">dom</a> <span class="p2">(<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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D2" title="CAT_1:def.2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c8">g'</font>
<b>by </b><i><a class="ref" href="modcat_1.html#D11" target="_self" title="MODCAT_1:def.11">Def11</a></i>
<br/>.= 
 <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c8">g'</font>

;<br/>
<i><font color="Green" title="E24">A2</font></i>:  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="modcat_1.html#K9" title="MODCAT_1:func.9">cod</a> <span class="p2">(<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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D3" title="CAT_1:def.3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="ref" href="modcat_1.html#D12" target="_self" title="MODCAT_1:def.12">Def12</a></i>
<br/>.= 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c7">f'</font>

;<br/>
<div><i><font color="Green" title="E25">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font></span><br/><b>then </b> <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c8">g'</font> = 
 <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="ref" href="modcat_1.html#D14" target="_self" title="MODCAT_1:def.14">Def14</a></i>
<br/>.= 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c7">f'</font>

;<br/><b>hence </b><a NAME="E3:32_3"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E3:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="modcat_1.html#E4:32"><i><font color="Green" title="E24">A2</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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_4"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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></span><br/><a NAME="E2:32_4"/><b>then </b>
 <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Maroon" title="c7">f'</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E3:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="modcat_1.html#E4:32"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:32_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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>
 <b>by </b><i><a class="ref" href="modcat_1.html#D14" target="_self" title="MODCAT_1:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:32"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</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> iff  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E5:32"><i><font color="Green" title="E25">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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