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<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</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">o1</font>,<font color="Maroon">o2</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">A</font> is <a href="altcat_3.html#V2" target="_self">coretraction</a> &amp; <font color="Maroon">A</font> is <a href="altcat_3.html#V5" target="_self">epi</a> )
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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">o2</font>,<font color="Maroon">o1</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="E3:31"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o2</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/>
<b>consider </b><font color="Maroon">B</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o1</font><b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> <a href="altcat_3.html#R1" target="_self">is_left_inverse_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:31"/>
<font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</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">o1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E5:31"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<a NAME="E7:31"/><b>then </b>
<font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E3:31"><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="E8:31"/><b>then </b>
<font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E3:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>;<br/>
<a NAME="E9:31"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E3:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>;<br/>
<a NAME="E10:31"/><b>then </b>
<font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:31"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">B</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:31"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> is <a href="altcat_3.html#V1" target="_self">retraction</a>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E41</font></i>: <font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span> = 
<font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="txt" href="altcat_3.html#E3:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E5:31"><i><font color="Green">E27</font></i></a>, , </i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font>
<b>by </b><i>, , <a class="txt" href="altcat_3.html#E9:31"><i><font color="Green">E28</font></i></a>, </i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font> = 
<font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="txt" href="altcat_3.html#E3:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E5:31"><i><font color="Green">E27</font></i></a>, , , </i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E5:31"><i><font color="Green">E27</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E15:31"/>
<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#E11:31"><i><font color="Green">E31</font></i></a>, </i>;<br/>


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