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

<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">o1</font> is <a href="altcat_3.html#V6">initial</a>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">o2</font> is <a href="altcat_3.html#V7">terminal</a>
 ;<br/>

<b>assume </b><a NAME="E3:15"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:15"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font><b> such that </b><br/><a NAME="E8:15"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E9:15"/>
for <font color="Olive">M1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>  st <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Olive">M1</font>
 <b>by </b><i>, <a class="ref" href="altcat_3.html#T25">ALTCAT_3:25</a></i>;<br/>
<a NAME="E10:15"/>
for <font color="Olive">o3</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>  ex <font color="Olive">M</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Olive">o3</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Olive">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( for <font color="Olive">M1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Olive">o3</font>  st <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Olive">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Olive">M1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">M</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o3</font><b> such that </b><br/><a NAME="E2:15_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( for <font color="Olive">M1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o3</font>  st <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Olive">M1</font> ) )
 <b>by </b><i>, <a class="ref" href="altcat_3.html#T25">ALTCAT_3:25</a></i>;<br/>
<b>take </b>
<font color="Maroon">M</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font>
;<br/>

<a NAME="E3:15_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></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="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">M</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon">M1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/>

<b>assume </b><a NAME="E5:15_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">i2</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o2</font><b> such that </b><br/><a NAME="E7:15_1"/>
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E8:15_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">M1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o2</font>  st <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Olive">M1</font>
 <b>by </b><i>, <a class="ref" href="altcat_3.html#T27">ALTCAT_3:27</a></i>;<br/>
<b>thus </b><font color="Maroon">M</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">l</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">i2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">M1</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:15"/>
<font color="Maroon">o2</font> is <a href="altcat_3.html#V6">initial</a>
 <b>by </b><i><a class="ref" href="altcat_3.html#T25">ALTCAT_3:25</a></i>;<br/>

<a NAME="E12:15"/>
for <font color="Olive">o3</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>  ex <font color="Olive">M</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o3</font>,<font color="Maroon">o1</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o3</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( for <font color="Olive">M1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o3</font>,<font color="Maroon">o1</font>  st <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o3</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Olive">M1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">M</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o3</font>,<font color="Maroon">o2</font><b> such that </b><br/><a NAME="E2:15_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o3</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( for <font color="Olive">M1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o3</font>,<font color="Maroon">o2</font>  st <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o3</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Olive">M1</font> ) )
 <b>by </b><i>, <a class="ref" href="altcat_3.html#T27">ALTCAT_3:27</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M</font>
;<br/>

<a NAME="E3:15_2"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o3</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></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="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:15_2"/>
<font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o3</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon">M1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o3</font>,<font color="Maroon">o1</font>;<br/>

<b>assume </b><a NAME="E5:15_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o3</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">i1</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o1</font><b> such that </b><br/><a NAME="E7:15_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E8:15_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">M1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o1</font>  st <font color="Olive">M1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">M1</font>
 <b>by </b><i>, <a class="ref" href="altcat_3.html#T25">ALTCAT_3:25</a></i>;<br/>
<b>thus </b><font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M</font> = 
<font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">l</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M1</font>
<b>by </b><i>, , , <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">i1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o1</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">M1</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:15"/>
<font color="Maroon">o1</font> is <a href="altcat_3.html#V7">terminal</a>
 <b>by </b><i><a class="ref" href="altcat_3.html#T27">ALTCAT_3:27</a></i>;<br/>


</div>
