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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>



<b>thus </b><a NAME="E1:19"/>
( <font color="Maroon">c</font> is <a href="cat_1.html#V7">initial</a> implies <font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a>  is <a href="cat_1.html#V6">terminal</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c</font> is <a href="cat_1.html#V7">initial</a>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2" target="_self">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D15">CAT_1:def 15</a><br/>

<a NAME="E2:19_1"/><i><font color="Green">E25</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:19_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">b</font></span>)</span> <a href="oppcat_1.html#K3" target="_self">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
;<br/>
<b>hence </b><a NAME="E6:19_1"/><i><font color="Green">E30</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>, <a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">b</font><b> such that </b><br/><a NAME="E8:19_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">g</font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>, <a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">b</font> holds  <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a>  as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a>  <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">f'</font>
;<br/>

<b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> ;<br/>

<a NAME="E10:19_1"/>
 <a href="oppcat_1.html#K4" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
;<br/>
<a NAME="E11:19_1"/><b>then </b>
 <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">g</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>, <a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:19_1"/>
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a>  is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D16">CAT_1:def 16</a><br/>

<a NAME="E3:19"/><i><font color="Green">E46</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E5:19"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E48</font></i>: 
(  <a href="oppcat_1.html#K4" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="oppcat_1.html#K4" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
;<br/>
<b>hence </b><a NAME="E7:19"/><i><font color="Green">E49</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> <b> such that </b><br/><a NAME="E9:19"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">g</font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a>  holds  <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> =  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">f</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">b</font> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">f'</font>
;<br/>

<b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E11:19"/>
<font color="Maroon">g</font> <a href="oppcat_1.html#K5" target="_self">opp</a>  is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b</font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c</font> <a href="oppcat_1.html#K3" target="_self">opp</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:19"/>
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>


</div>
