<?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>assume </b><a NAME="E1:34"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">A1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o2</font> holds  <font color="Olive">A1</font> is <a href="altcat_3.html#V1" target="_self">retraction</a>
 ;<br/>

<b>thus </b><a NAME="E2:34"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">A</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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="Olive">b</font>,<font color="Olive">a</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>  holds <br/><font color="Olive">A</font> is <a href="altcat_3.html#V3" target="_self">iso</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>



<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">b</font>,<font color="Maroon">a</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/>

<a NAME="E2:34_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> is <a href="altcat_3.html#V1" target="_self">retraction</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_1"/>
<font color="Maroon">A</font> is <a href="altcat_3.html#V2" target="_self">coretraction</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">A1</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font><b> such that </b><br/><a NAME="E2:34_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A1</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E3:34_1_1"/>
<font color="Maroon">A1</font> is <a href="altcat_3.html#V1" target="_self">retraction</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:34_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">A1</font> is <a href="altcat_3.html#V5" target="_self">epi</a>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:34_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="altcat_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:34_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</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#T23">ALTCAT_1:23</a></i>;<br/>
<a NAME="E6:34_1_1"/>
<font color="Maroon">A1</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</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">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:34_1_1"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<a NAME="E7:34_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:34_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>;<br/>
<a NAME="E8:34_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:34_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>;<br/>
<a NAME="E9:34_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:34_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>;<br/>
<a NAME="E10:34_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E4:34_1_1"><i><font color="Green">E28</font></i></a>, , </i>;<br/>
<a NAME="E11:34_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="altcat_3.html#R1" target="_self">is_left_inverse_of</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:34_1_1"/>
<font color="Maroon">A</font> is <a href="altcat_3.html#V2" target="_self">coretraction</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_1"/>
<font color="Maroon">A</font> is <a href="altcat_3.html#V3" target="_self">iso</a>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:34_1"><i><font color="Green">E25</font></i></a>, , </i>;<br/>


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


</div>
