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

<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>assume </b><a NAME="E1:9"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> holds  <font color="Olive" title="b3">f</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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>let </b><font color="Maroon" title="c4">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>



<b>assume </b><a NAME="E2:9"/><i><font color="Green" title="E6">A2</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">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c2">a</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>  )
 ;<br/>

<a NAME="E3:9"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:9"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<a NAME="E4:9"/>
<font color="Maroon" title="c4">g</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E2:9_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E3:9"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="altcat_3.html#D3" title="ALTCAT_3:def.3">ALTCAT_3:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D2" title="ALTCAT_3:def.2">ALTCAT_3:def 2</a><br/>

<a NAME="E3:9_1"/>
<font color="Maroon" title="c5">f</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:9"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<a NAME="E4:9_1"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="altcat_3.html#T16" title="ALTCAT_3:th.16">ALTCAT_3:16</a></i>;<br/>
<font color="Maroon" title="c5">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font>
<b>by </b><i><a class="txt" href="altcat_4.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font>
<b>by </b><i><a class="txt" href="altcat_4.html#E2:9_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font>
<b>by </b><i><a class="txt" href="altcat_4.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="altcat_1.html#T24" title="ALTCAT_1:th.24">ALTCAT_1:24</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_4.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
;<br/>
<b>hence </b><a NAME="E6:9_1"/>
<font color="Maroon" title="c4">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> <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">b</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E4:9_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="altcat_3.html#D7" title="ALTCAT_3:def.7">ALTCAT_3:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:9"/>
<font color="Maroon" title="c4">g</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#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="altcat_4.html#E3:9"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="altcat_3.html#T6" title="ALTCAT_3:th.6">ALTCAT_3:6</a></i>;<br/>


</div>
