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

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

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E17</font></i>: 
 <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> #) <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> #)
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:26"/><i><font color="Green">E40</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> is <a href="cat_5.html#V2" target="_self">categorial</a>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">A</font> being  <a href="cat_1.html#NM4">Category</a>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Olive">a</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="Olive">A</font>,<font color="Olive">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="Olive">A</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">m</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="cat_1.html#NM4">Category</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Olive">m</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">m</font> holds <br/> ex <font color="Olive">F</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">A</font>,<font color="Olive">B</font> st <font color="Olive">m</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="Olive">A</font>,<font color="Olive">B</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">F</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:26"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">m1</font>, <font color="Olive">m2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><br/> for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="cat_1.html#NM4">Category</a><br/> for <font color="Olive">F</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">A</font>,<font color="Olive">B</font><br/> for <font color="Olive">G</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">B</font>,<font color="Olive">C</font>  st <font color="Olive">m1</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="Olive">A</font>,<font color="Olive">B</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">F</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">m2</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="Olive">B</font>,<font color="Olive">C</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">G</font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Olive">m2</font> <a href="cat_1.html#K4">*</a> <font color="Olive">m1</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="Olive">A</font>,<font color="Olive">C</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">G</font> <a href="cat_1.html#K14">*</a> <font color="Olive">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D6" target="_self">CAT_5:def 6</a><br/>

<b>thus </b><a NAME="E6:26"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font> is <a href="cat_5.html#V2" target="_self">categorial</a>
 <b>by </b><i>, <a class="txt" href="cat_5.html#E1:26"><i><font color="Green">E17</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D6" target="_self">CAT_5:def 6</a><br/>

<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">D</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:26_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/><b>then </b><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> = 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>

;<br/><b>hence </b><a NAME="E4:26_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a</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">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/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font>;<br/><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>assume </b><a NAME="E2:26_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E3:26_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E4:26_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m'</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:26_2"/>
 ex <font color="Olive">F</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> st <font color="Maroon">m</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="Olive">F</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">m1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font>, <font color="Maroon">m2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font>;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">m1</font>, <font color="Maroon">f2</font> = <font color="Maroon">m2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">c</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/>





<b>assume </b><a NAME="E10:26"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">m1</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> &amp; <font color="Maroon">m2</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">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">G</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">a1</font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font>, <font color="Maroon">b1</font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font>, <font color="Maroon">a2</font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font>, <font color="Maroon">b2</font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> as   <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="txt" href="cat_5.html#E1:26"><i><font color="Green">E17</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">a1</font>,<font color="Maroon">b1</font><b> such that </b><br/><a NAME="E13:26"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f1</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">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">a2</font>,<font color="Maroon">b2</font><b> such that </b><br/><a NAME="E15:26"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f2</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">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E16:26"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> <a href="mcart_1.html#K17">`11</a>  &amp; <font color="Maroon">m2</font> <a href="mcart_1.html#K17">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="mcart_1.html#K18">`12</a>  &amp; <font color="Maroon">m1</font> <a href="mcart_1.html#K18">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="cat_5.html#D1" target="_self">Def1</a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/>
<a NAME="E17:26"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f2</font>,<font color="Maroon">f1</font></span><a href="cat_2.html#K13">]</a></span> <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>
 
<b>by </b><i><a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>
<b>hence </b><font color="Maroon">m2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">m1</font> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f1</font>
<b>by </b><i>, <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"><font color="Maroon">a</font>,<font color="Maroon">c</font></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/><br/>


</div>
