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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a> <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">F</font> be  <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a> <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>  <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c4">G</font> be  <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a> <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>  <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>;<br/>





<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> &amp; <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">a2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:3"/><i><font color="Green" title="E5">A2</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c7">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">a2</font>;<br/><b>let </b><font color="Maroon" title="c8">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a1</font></span>)</span>;<br/>



<a NAME="E3:3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a> </span>)</span> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="functor1.html#T22" title="FUNCTOR1:th.22">FUNCTOR1:22</a></i>;<br/>
<a NAME="E4:3"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="yellow20.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="functor0.html#T32" title="FUNCTOR0:th.32">FUNCTOR0:32</a></i>;<br/>
<b>hence </b><a NAME="E5:3"/>
( <font color="Maroon" title="c3">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> implies <font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="functor3.html#T7" title="FUNCTOR3:th.7">FUNCTOR3:7</a></i>;<br/>

<a NAME="E6:3"/><i><font color="Green" title="E6">A3</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a1</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="yellow20.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="functor0.html#D20" title="FUNCTOR0:def.20">FUNCTOR0:def 20</a></i>;<br/>
<a NAME="E7:3"/>
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V20" title="FUNCTOR0:attr.20">surjective</a>
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="functor0.html#D36" title="FUNCTOR0:def.36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E8:3"/><b>then </b>
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a></i>;<br/>
<a NAME="E9:3"/><b>then </b>
( <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V6" title="FUNCTOR0:attr.6">reflexive</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V7" title="FUNCTOR0:attr.7">coreflexive</a> )
 
<b>by </b><i><a class="ref" href="functor0.html#T46" title="FUNCTOR0:th.46">FUNCTOR0:46</a></i>;<br/>
<a NAME="E10:3"/><b>then </b><i><font color="Green" title="E7">A4</font></i>: 
( <font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a1</font> &amp; <font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a2</font> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="yellow20.html#T1" target="_self" title="YELLOW20:th.1">Th1</a></i>;<br/>
<a NAME="E11:3"/>
<font color="Maroon" title="c3">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="functor1.html#T21" title="FUNCTOR1:th.21">FUNCTOR1:21</a></i>;<br/>
<a NAME="E12:3"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">g</font>
 
<b>by </b><i><a class="txt" href="yellow20.html#E6:3"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="functor0.html#T32" title="FUNCTOR0:th.32">FUNCTOR0:32</a></i>;<br/>
<b>hence </b><a NAME="E13:3"/>
( <font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> implies <font color="Maroon" title="c3">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E6:3"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="yellow20.html#E10:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="functor3.html#T7" title="FUNCTOR3:th.7">FUNCTOR3:7</a></i>;<br/>


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