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

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

<b>set </b><font color="Maroon">M</font> = the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">d</font> = the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">c</font> = the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">i</font> = the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:2_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> iff the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:2_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">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="funct_4.html#T43">FUNCT_4:43</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> iff the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:2_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/><span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:2_1_2"/>
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:2_1_2"/>
<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T44">FUNCT_4:44</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:2_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:2_1_3"/>
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> implies <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:2_1_3"/>
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> implies ( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> ) )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:2_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/><span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">h</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:2_1_4"/><i><font color="Green">E28</font></i>: 
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:2_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:2_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> is    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> is    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> )
 
<b>by </b><i>, , , <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E4:2_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:2_1_4"/>
<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>;<br/><b>thus </b><a NAME="E5:2_1"/><i><font color="Green">E30</font></i>: 
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/><b>thus </b><a NAME="E6:2_1"/>
for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:2_1_5"/>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default">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></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:2_1_5"/>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/><a NAME="E7:2_1"/>
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default">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></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">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></span>)</span>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:2_1"/>
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default">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></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:2"/>
 <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#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> #) is   <a href="cat_1.html#NM4">Category</a>
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div>
