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<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><a NAME="E1:47"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="yellow18.html#R2" target="_self">are_opposite</a> 
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">b</font>,<font color="Maroon">a</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#NM1">object</a> of <font color="Maroon">B</font>, <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E3:47"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E4:47"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">a'</font>;<br/>



<b>assume </b><a NAME="E5:47"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> is <a href="altcat_3.html#V1">retraction</a> &amp; <font color="Maroon">f</font> is <a href="altcat_3.html#V2">coretraction</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a>  as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E7:47"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="altcat_3.html#T2">ALTCAT_3:2</a></i>;<br/>
<a NAME="E8:47"/><b>then </b>
( <font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E9:47"/><b>then </b>
( <font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b'</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:47"/><b>then </b>
( <font color="Maroon">f'</font> is <a href="altcat_3.html#V1">retraction</a> &amp; <font color="Maroon">f'</font> is <a href="altcat_3.html#V2">coretraction</a> &amp; <font color="Maroon">g</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">g</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">f'</font> )
 
<b>by </b><i>, , , , , <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:47"/>
<font color="Maroon">f'</font> <a href="altcat_3.html#K1">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a> 
 <b>by </b><i>, , <a class="ref" href="altcat_3.html#D4">ALTCAT_3:def 4</a></i>;<br/>


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