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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">C</font> is <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a>
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font> is <a href="cat_5.html#V2" title="CAT_5:attr.2">categorial</a> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b3">B</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">m</font> holds <br/> ex <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> st <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 
<b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D1" target="_self" title="CAT_5:def.1">CAT_5:def 1</a><br/>

<b>reconsider </b><font color="Maroon" title="c3">A</font> =  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c4">B</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font> as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <b>by </b><i><a class="txt" href="cat_5.html#E2:25_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="cat_5.html#D5" target="_self" title="CAT_5:def.5">Def5</a></i>;<br/>
<a NAME="E4:25_1_1"/>
 ex <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> st <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b1">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:25_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>


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