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<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>thus </b><a NAME="E1:19"/>
( <font color="Maroon" title="c2">c</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a> implies <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  is <a href="cat_1.html#V6" title="CAT_1:attr.6">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" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">c</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D15" title="CAT_1:def.15">CAT_1:def 15</a><br/>

<a NAME="E2:19_1"/><i><font color="Green" title="E11">A2</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">c</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font></span>)</span> <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="txt" href="oppcat_1.html#E1:19_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="cat_1.html#D16" title="CAT_1:def.16">CAT_1:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">c</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:19_1"/><i><font color="Green" title="E13">A4</font></i>: 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 
;<br/>
<b>hence </b><a NAME="E6:19_1"/><i><font color="Green" title="E14">A5</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> <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="txt" href="oppcat_1.html#E4:19_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="oppcat_1.html#T13" target="_self" title="OPPCAT_1:th.13">Th13</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c5">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">c</font>, <a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font><b> such that </b><br/><a NAME="E8:19_1"/><i><font color="Green" title="E15">A6</font></i>: 
for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">c</font>, <a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font> holds  <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="oppcat_1.html#E1:19_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="cat_1.html#D16" title="CAT_1:def.16">CAT_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">f'</font> = <font color="Maroon" title="c5">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <b>by </b><i><a class="txt" href="oppcat_1.html#E2:19_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="oppcat_1.html#E5:19_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="oppcat_1.html#T15" target="_self" title="OPPCAT_1:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">f'</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ;<br/>

<a NAME="E10:19_1"/>
 <a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font>
 
;<br/>
<a NAME="E11:19_1"/><b>then </b>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c7">g</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">c</font>, <a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E6:19_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="oppcat_1.html#T16" target="_self" title="OPPCAT_1:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E12:19_1"/>
<font color="Maroon" title="c6">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="oppcat_1.html#E8:19_1"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:19"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  is <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D16" title="CAT_1:def.16">CAT_1:def 16</a><br/>

<a NAME="E3:19"/><i><font color="Green" title="E11">A8</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> <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="txt" href="oppcat_1.html#E2:19"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="cat_1.html#D15" title="CAT_1:def.15">CAT_1:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span><b> such that </b><br/><a NAME="E5:19"/><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E13">A10</font></i>: 
(  <a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
;<br/>
<b>hence </b><a NAME="E7:19"/><i><font color="Green" title="E14">A11</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">b</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="txt" href="oppcat_1.html#E5:19"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="oppcat_1.html#T14" target="_self" title="OPPCAT_1:th.14">Th14</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c5">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> <b> such that </b><br/><a NAME="E9:19"/><i><font color="Green" title="E15">A12</font></i>: 
for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  holds  <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="oppcat_1.html#E2:19"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="cat_1.html#D15" title="CAT_1:def.15">CAT_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">f'</font> =  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c5">f</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">b</font> <b>by </b><i><a class="txt" href="oppcat_1.html#E3:19"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="oppcat_1.html#E6:19"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="oppcat_1.html#T16" target="_self" title="OPPCAT_1:th.16">Th16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">f'</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">b</font>;<br/>

<a NAME="E11:19"/>
<font color="Maroon" title="c7">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E7:19"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="oppcat_1.html#T15" target="_self" title="OPPCAT_1:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E12:19"/>
<font color="Maroon" title="c6">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="oppcat_1.html#E9:19"><i><font color="Green" title="E15">A12</font></i></a></i>;<br/>


</div>
