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

<b>let </b><font color="Maroon">C</font> be  <a href="cat_5.html#V3" target="_self">Categorial</a> <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">D</font> be  <a href="cat_5.html#V3" target="_self">Categorial</a> <a href="cat_1.html#NM4">Category</a>;<br/>

<b>assume </b><a NAME="E1:34"/><i><font color="Green">E17</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>
 ;<br/>

<b>thus </b><a NAME="E2:34"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> ;<br/>
<b>reconsider </b><font color="Maroon">i</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:34_1"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="cat_5.html#K8" target="_self">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="cat_5.html#K8" target="_self">`11</a>  )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font>
 ;<br/>


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

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Maroon">b'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 ;<br/><b>thus </b><a NAME="E2:34_2"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:34_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> : (  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:34_2_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <font color="Maroon">B</font> = <font color="Maroon">b</font> as   <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="txt" href="cat_5.html#E1:34"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E7:34_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:34_2_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_5.html#K8" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_5.html#K9" target="_self">`12</a>  &amp; <font color="Maroon">f</font> <a href="cat_5.html#K8" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="cat_5.html#K9" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, , <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/>
<b>hence </b><a NAME="E10:34_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="cat_5.html#D1" target="_self">Def1</a></i>;<br/>


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

<a NAME="E4:34"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">f</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:34_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E5:34_3"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_5.html#K8" target="_self">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_5.html#K8" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_5.html#K9" target="_self">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_5.html#K9" target="_self">`12</a>  )
 
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, , <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E6:34_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 ;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">f</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:34_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E5:34_4"/><i><font color="Green">E54</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_5.html#K8" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_5.html#K9" target="_self">`12</a>  )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">G</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">g</font> <a href="cat_5.html#K8" target="_self">`11</a> ,<font color="Maroon">g</font> <a href="cat_5.html#K9" target="_self">`12</a> <b> such that </b><br/><a NAME="E7:34_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="cat_5.html#K8" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="cat_5.html#K9" target="_self">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">G</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E8:34_4"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_5.html#K8" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_5.html#K9" target="_self">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, , <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">f</font> <a href="cat_5.html#K8" target="_self">`11</a> ,<font color="Maroon">g</font> <a href="cat_5.html#K8" target="_self">`11</a> <b> such that </b><br/><a NAME="E10:34_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="cat_5.html#K8" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="cat_5.html#K8" target="_self">`11</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/><b>thus </b>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="cat_5.html#K8" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="cat_5.html#K9" target="_self">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">g'</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f'</font>
<b>by </b><i>, , </i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E7:34"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font> as   <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="txt" href="cat_5.html#E1:34"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>thus </b> <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, </i>
;<br/>


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