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<b>let </b><font color="Maroon">I</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/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><a NAME="E1:17_1"/>
( <font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a>  <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">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">~</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> #) &amp; <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span> <a href="oppcat_1.html#K2">opp</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K1">~</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span></span>)</span>,the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span> #) )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D1">OPPCAT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E2:17_1"/>
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span> <a href="oppcat_1.html#K2">opp</a> </span>)</span>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:17_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="oppcat_1.html#K5">opp</a> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="oppcat_1.html#T6">OPPCAT_1:6</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="cat_3.html#K5" target="_self">opp</a>  <a href="funct_2.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


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