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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V12">with_units</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:5"/>
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font> is <a href="altcat_3.html#V5">epi</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D8">ALTCAT_3:def 8</a><br/>

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E5</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> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <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>;<br/>

<b>assume </b><a NAME="E2:5_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:5_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:5_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="altcat_4.html#E2:5_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
;<br/>


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

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D7">ALTCAT_3:def 7</a><br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E5</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>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> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:5"/><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_4.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="altcat_4.html#E3:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
;<br/>


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