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<div class="add">

<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font> <a href="pboole.html#K9">--&gt;</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E3:80_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font> holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E4:80_1_1"/>
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>  ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">D</font> st <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E1:80_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:80_1_1"/>
for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font> holds <br/> ( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_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" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:80_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>


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

<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" target="_self">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" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E6:80_1_1"/><i><font color="Green">E22</font></i>: 
<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> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font>
 ;<br/>

<b>set </b><font color="Maroon">f'</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>;<br/>
<a NAME="E7:80_1_1"/>
( <font color="Maroon">F</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#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:80_1_1"/><b>then </b>
( the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:80_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></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" target="_self">Comp</a> of <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:80_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:80_1_1"/>
the <a href="cat_1.html#U5" target="_self">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">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" target="_self">Morphisms</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <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> = 
the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d</font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">d</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">D</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"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
;<br/><br/>


</div>
