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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="altcat_3.html#V6">initial</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="altcat_3.html#R2">are_iso</a> 
 ;<br/>

<a NAME="E3:14"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E2:14_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E3:14_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><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="altcat_4.html#E1:14"><i><font color="Green">E5</font></i></a>, <a class="ref" href="altcat_3.html#T25">ALTCAT_3:25</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:14_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="txt" href="altcat_4.html#E2:14"><i><font color="Green">E6</font></i></a>, <a class="ref" href="altcat_3.html#D6">ALTCAT_3:def 6</a></i>;<br/>
<a NAME="E6:14_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E5:14_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span>
;<br/>

<a NAME="E7:14_1"/><i><font color="Green">E11</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></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="txt" href="altcat_4.html#E2:14"><i><font color="Green">E6</font></i></a>, <a class="ref" href="altcat_3.html#D6">ALTCAT_3:def 6</a></i>;<br/>
<a NAME="E8:14_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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="txt" href="altcat_4.html#E2:14_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:14_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>

<b>assume </b><a NAME="E10:14_1"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<a NAME="E11:14_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E2:14_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="altcat_4.html#E3:14_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:14_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E6:14_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="altcat_4.html#E7:14_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="altcat_4.html#E8:14_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="altcat_4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">c<sub>3</sub></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/>


</div>
