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

<b>let </b><font color="Maroon" title="c2">D</font> be    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font>;<br/>

<a NAME="E1:33_1_1"/><i><font color="Green" title="E20">A1</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> ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</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="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b6">f</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font><br/> for <font color="Olive" title="b7">g</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>  st <font color="Olive" title="b1">m1</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="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">m2</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="b4">b</font>,<font color="Olive" title="b5">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">m1</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="b3">a</font>,<font color="Olive" title="b5">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">g</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b6">f</font></span>)</span></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>thus </b><a NAME="E2:33_1_1"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">D</font> is <a href="cat_5.html#V2" title="CAT_5:attr.2">categorial</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">CAT_5:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D5" target="_self" title="CAT_5:def.5">CAT_5:def 5</a><br/>

<a NAME="E1:33_1_1_1"/>
<font color="Maroon" title="c3">x</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="cat_2.html#T12" title="CAT_2:th.12">CAT_2:12</a></i>;<br/>
<b>hence </b><a NAME="E2:33_1_1_1"/>
<font color="Maroon" title="c3">x</font> is   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:33_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="cat_5.html#D4" target="_self" title="CAT_5:def.4">Def4</a></i>;<br/>


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

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c3">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="cat_2.html#T12" title="CAT_2:th.12">CAT_2:12</a></i>;<br/><b>assume </b><a NAME="E2:33_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 ;<br/><a NAME="E3:33_1_1_2"/><b>then </b>
<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="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c4">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_2"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</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="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c4">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font>;<br/><b>reconsider </b><font color="Maroon" title="c4">m'</font> = <font color="Maroon" title="c3">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="cat_2.html#T14" title="CAT_2:th.14">CAT_2:14</a></i>;<br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>assume </b><a NAME="E2:33_1_1_3"/>
( <font color="Maroon" title="c5">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="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c6">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="Maroon" title="c3">m</font> )
 ;<br/><a NAME="E3:33_1_1_3"/><b>then </b>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="cat_2.html#T15" title="CAT_2:th.15">CAT_2:15</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_3"/>
 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="c5">a</font>,<font color="Maroon" title="c6">b</font> st <font color="Maroon" title="c3">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c4">m2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>



<b>reconsider </b><font color="Maroon" title="c8">m1'</font> = <font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c9">m2'</font> = <font color="Maroon" title="c4">m2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="cat_2.html#T14" title="CAT_2:th.14">CAT_2:14</a></i>;<br/>
<b>let </b><font color="Maroon" title="c10">f</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>;<br/><b>let </b><font color="Maroon" title="c11">g</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>;<br/>



<b>assume </b><a NAME="E6:33_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c3">m1</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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">m2</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="c6">b</font>,<font color="Maroon" title="c7">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<a NAME="E7:33_1_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m2</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="Maroon" title="c9">m2'</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">m1</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="c8">m1'</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c9">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m2</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c8">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m1</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 
<b>by </b><i><a class="ref" href="cat_5.html#T13" target="_self" title="CAT_5:th.13">Th13</a>, <a class="ref" href="cat_2.html#T15" title="CAT_2:th.15">CAT_2:15</a></i>;<br/>
<a NAME="E8:33_1_1"/><b>then </b>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E6:33_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c3">m1</font> = 
<font color="Maroon" title="c9">m2'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c8">m1'</font>
<b>by </b><i><a class="ref" href="cat_2.html#T17" title="CAT_2:th.17">CAT_2:17</a></i>
<br/>.= 
<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="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c10">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E6:33_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>
;<br/><br/>


</div>
