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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>assume </b><a NAME="E1:8"/><i><font color="Green">E21</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/>

<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><a NAME="E2:8"/>
( <font color="Maroon">A</font> is <a href="altcat_3.html#V1" target="_self">retraction</a> &amp; <font color="Maroon">A</font> is <a href="altcat_3.html#V2" target="_self">coretraction</a> )
 ;<br/>

<a NAME="E3:8"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a>  <a href="altcat_3.html#R1" target="_self">is_left_inverse_of</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a>  <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="E4:8"/><b>then </b>
( <font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a>  is <a href="altcat_3.html#V1" target="_self">retraction</a> &amp; <font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a>  is <a href="altcat_3.html#V2" target="_self">coretraction</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green">E25</font></i>: 
<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_3.html#K1" target="_self">"</a>  <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a> 
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><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_3.html#K1" target="_self">"</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">A</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span> <a href="altcat_3.html#K1" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<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>, </i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
;<br/>


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