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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:41"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V5">invertible</a> implies ( <font color="Maroon">c<sub>2</sub></font> is <a href="cat_3.html#V1" target="_self">retraction</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="cat_3.html#V2" target="_self">coretraction</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12">CAT_1:def 12</a><br/>

<b>hence </b><a NAME="E2:41_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="cat_3.html#V1" target="_self">retraction</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="cat_3.html#V2" target="_self">coretraction</a> )
 <b>by </b><i><a class="ref" href="cat_3.html#D10" target="_self">Def10</a>, <a class="ref" href="cat_3.html#D11" target="_self">Def11</a></i>;<br/>


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

<b>assume </b><a NAME="E2:41"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="cat_3.html#V1" target="_self">retraction</a>
 ;<br/>

<a NAME="E3:41"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V4">epi</a>
 
<b>by </b><i><a class="ref" href="cat_3.html#T26" target="_self">Th26</a></i>;<br/>
<b>hence </b><a NAME="E4:41"/>
(  not <font color="Maroon">c<sub>2</sub></font> is <a href="cat_3.html#V2" target="_self">coretraction</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V5">invertible</a> )
 <b>by </b><i><a class="ref" href="cat_3.html#T33" target="_self">Th33</a></i>;<br/>


</div>
