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<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V12">with_units</a>  <a href="altcat_1.html#L2">AltCatStr</a> , <font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</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">F</font> be  <a href="functor0.html#V8">feasible</a> <a href="functor0.html#V10">Covariant</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G</font> be  <a href="functor0.html#V8">feasible</a> <a href="functor0.html#V10">Covariant</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">F</font> is <a href="functor0.html#V21">bijective</a> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K15">"</a>  )
 ;<br/>

<b>let </b><font color="Maroon">a1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:2"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</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">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a2</font></span>)</span>;<br/>



<a NAME="E3:2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K15">"</a> </span>)</span> <a href="functor0.html#K13">*</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K12">id</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="functor1.html#T22">FUNCTOR1:22</a></i>;<br/>
<a NAME="E4:2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor0.html#K13">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="functor0.html#T32">FUNCTOR0:32</a></i>;<br/>
<b>hence </b><a NAME="E5:2"/>
( <font color="Maroon">F</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> implies <font color="Maroon">G</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="functor3.html#T6">FUNCTOR3:6</a></i>;<br/>

<a NAME="E6:2"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a2</font></span>)</span></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="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<a NAME="E7:2"/>
<font color="Maroon">F</font> is <a href="functor0.html#V20">surjective</a>
 
<b>by </b><i>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E8:2"/><b>then </b>
<font color="Maroon">F</font> is <a href="functor0.html#V5">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a></i>;<br/>
<a NAME="E9:2"/><b>then </b>
( <font color="Maroon">F</font> is <a href="functor0.html#V6">reflexive</a> &amp; <font color="Maroon">F</font> is <a href="functor0.html#V7">coreflexive</a> )
 
<b>by </b><i><a class="ref" href="functor0.html#T45">FUNCTOR0:45</a></i>;<br/>
<a NAME="E10:2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:2"/>
<font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K12">id</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="functor1.html#T21">FUNCTOR1:21</a></i>;<br/>
<a NAME="E12:2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="functor0.html#T32">FUNCTOR0:32</a></i>;<br/>
<b>hence </b><a NAME="E13:2"/>
( <font color="Maroon">G</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> implies <font color="Maroon">F</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , <a class="ref" href="functor3.html#T6">FUNCTOR3:6</a></i>;<br/>


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