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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:20"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a>  is <a href="cat_1.html#V7">initial</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V6">terminal</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a>  is <a href="cat_1.html#V7">initial</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D15">CAT_1:def 15</a><br/>

<a NAME="E2:20_1"/><i><font color="Green">E11</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="txt" href="oppcat_1.html#E1:20_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="oppcat_1.html#K2" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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="E5:20_1"/><i><font color="Green">E13</font></i>: 
(  <a href="oppcat_1.html#K4" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="oppcat_1.html#K4" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E6:20_1"/><i><font color="Green">E14</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="oppcat_1.html#E4:20_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="oppcat_1.html#T14" target="_self">Th14</a></i>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> <b> such that </b><br/><a NAME="E8:20_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a>  holds <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="oppcat_1.html#E1:20_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="oppcat_1.html#E2:20_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="oppcat_1.html#E5:20_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="oppcat_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E10:20_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a>  is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> 
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E6:20_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="oppcat_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> = 
 <a href="oppcat_1.html#K6" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="oppcat_1.html#E8:20_1"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>

;<br/><br/>


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

<b>assume </b><a NAME="E2:20"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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#D16">CAT_1:def 16</a><br/>

<a NAME="E3:20"/><i><font color="Green">E11</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E2:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:20"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="oppcat_1.html#K3" target="_self">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>hence </b><a NAME="E7:20"/><i><font color="Green">E14</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="oppcat_1.html#E5:20"><i><font color="Green">E12</font></i></a>, <a class="ref" href="oppcat_1.html#T13" target="_self">Th13</a></i>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of  <a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:20"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of  <a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="oppcat_1.html#E2:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#D15">CAT_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a>  as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="oppcat_1.html#E3:20"><i><font color="Green">E11</font></i></a>, <a class="txt" href="oppcat_1.html#E6:20"><i><font color="Green">E13</font></i></a>, <a class="ref" href="oppcat_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> ,<font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E11:20"/>
 <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>7</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of  <a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font>, <a href="oppcat_1.html#K4" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K3" target="_self">opp</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E7:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="oppcat_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
 <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>7</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of  <a href="oppcat_1.html#K4" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 
<b>by </b><i><a class="txt" href="oppcat_1.html#E9:20"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>

;<br/><br/>


</div>
