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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="Red">::  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">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="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>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">f</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="Red">::  thesis: </font></span></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="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>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</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">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">f9</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">g9</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/>
<span class="lab"><font color="Green" title="E23">A1</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> = 
 <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Maroon" title="c7">f9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D12" target="_self" title="MODCAT_1:def.12">Def12</a></span>
<br/>.= 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c7">f9</font>

;<br/>
<span class="lab"><font color="Green" title="E24">A2</font></span>:  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> = 
 <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c8">g9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D11" target="_self" title="MODCAT_1:def.11">Def11</a></span>
<br/>.= 
 <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c8">g9</font>

;<br/>
<div><span class="lab"><font color="Green" title="E25">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_3"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:34_3"/><span class="kw">then </span>
 <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c8">g9</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">f9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D14" target="_self" title="MODCAT_1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font></span><br/><span class="kw">then </span> <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c8">g9</font> = 
 <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Maroon" title="c7">f9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D14" target="_self" title="MODCAT_1:def.14">Def14</a></span>
<br/>.= 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c7">f9</font>

;<br/><span class="kw">hence </span><a NAME="E3:34_4"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:34"/>
( <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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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