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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/><b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">o1</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font> <a href="altcat_3.html#R2" title="ALTCAT_3:pred.2">are_iso</a> 
 ;<br/>

<a NAME="E3:14"/>
for <font color="Olive" title="b1">o3</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">M</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Olive" title="b1">o3</font> st <br/>( <font color="Olive" title="b2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Olive" title="b1">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; ( for <font color="Olive" title="b3">v</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Olive" title="b1">o3</font>  st <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Olive" title="b1">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</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" title="c4">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>consider </b><font color="Maroon" title="c5">u</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font><b> such that </b><br/><a NAME="E2:14_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E3:14_1"/><i><font color="Green" title="E8">A4</font></i>: 
for <font color="Olive" title="b1">M1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font>  st <font color="Olive" title="b1">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M1</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:14"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="altcat_3.html#T25" title="ALTCAT_3:th.25">ALTCAT_3:25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font><b> such that </b><br/><a NAME="E5:14_1"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="altcat_4.html#E2:14"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="altcat_3.html#D6" title="ALTCAT_3:def.6">ALTCAT_3:def 6</a></i>;<br/>
<a NAME="E6:14_1"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E5:14_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="altcat_3.html#D5" title="ALTCAT_3:def.5">ALTCAT_3:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">u</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span>
;<br/>

<a NAME="E7:14_1"/><i><font color="Green" title="E11">A7</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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="altcat_4.html#E2:14"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="altcat_3.html#D6" title="ALTCAT_3:def.6">ALTCAT_3:def 6</a></i>;<br/>
<a NAME="E8:14_1"/><b>then </b><i><font color="Green" title="E12">A8</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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="altcat_4.html#E2:14_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:14_1"/>
<font color="Maroon" title="c5">u</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">v</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>;<br/>

<b>assume </b><a NAME="E10:14_1"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/>

<a NAME="E11:14_1"/>
<font color="Maroon" title="c7">v</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E2:14_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="altcat_4.html#E3:14_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:14_1"/>
<font color="Maroon" title="c5">u</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E6:14_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="altcat_4.html#E7:14_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="altcat_4.html#E8:14_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="altcat_4.html#T2" target="_self" title="ALTCAT_4:th.2">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon" title="c3">o2</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a>
 <b>by </b><i><a class="ref" href="altcat_3.html#T25" title="ALTCAT_3:th.25">ALTCAT_3:25</a></i>;<br/>


</div>
