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

<b>consider </b><font color="Maroon" title="c3">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">D</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">F</font> = the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">F</font>
;<br/>

<b>thus </b><a NAME="E3:79_1_1"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">D</font> st <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>take </b>
<font color="Maroon" title="c3">d</font>
;<br/>

<b>thus </b><a NAME="E1:79_1_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>


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

<b>thus </b><a NAME="E4:79_1_1"/>
for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>;<br/>

<a NAME="E1:79_1_1_2"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> = 
the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:79_1_1_2"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="cat_1.html#D8" target="_self" title="CAT_1:def.8">Def8</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> = 
the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:79_1_1_2"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="cat_1.html#D8" target="_self" title="CAT_1:def.8">Def8</a></i>
;<br/>


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

<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><a NAME="E5:79_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font>;<br/>
<a NAME="E6:79_1_1"/>
( <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:79_1_1"/><b>then </b>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#D8" target="_self" title="CAT_1:def.8">Def8</a></i>;<br/>
<a NAME="E8:79_1_1"/><b>then </b><i><font color="Green" title="E39">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#D8" target="_self" title="CAT_1:def.8">Def8</a></i>;<br/>
<a NAME="E9:79_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T55" target="_self" title="CAT_1:th.55">Th55</a></i>;<br/>
<a NAME="E10:79_1_1"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="cat_1.html#E5:79_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font></span>)</span> = 
the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">d</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59" target="_self" title="CAT_1:th.59">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">d</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E9:79_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="cat_1.html#D13" target="_self" title="CAT_1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E8:79_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="cat_1.html#D4" target="_self" title="CAT_1:def.4">Def4</a></i>
;<br/><br/>


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