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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>



<a NAME="E1:18"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="index_1.html#K3" target="_self">Objs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="index_1.html#K4" target="_self">Mphs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="index_1.html#K3" target="_self">Objs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:18_1"/>
<span class="p1">(<span class="default"><a href="index_1.html#K3" target="_self">Objs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
 <a href="index_1.html#K3" target="_self">Objs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="index_1.html#K4" target="_self">Mphs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:18_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:18_2"/>
<span class="p1">(<span class="default"><a href="index_1.html#K4" target="_self">Mphs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
 <a href="index_1.html#K4" target="_self">Mphs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>


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