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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  implies ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a>  iff <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R2" title="FUNCTOR0:pred.2">are_anti-isomorphic</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:40"/>
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a>  iff <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R2" title="FUNCTOR0:pred.2">are_anti-isomorphic</a>  )</span><br/>

<a NAME="E2:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow18.html#T15" target="_self" title="YELLOW18:th.15">Th16</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R2" title="FUNCTOR0:pred.2">are_anti-isomorphic</a>  implies <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:40_1"/>
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R2" title="FUNCTOR0:pred.2">are_anti-isomorphic</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">F</font> being    <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:40_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
( <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D39" title="FUNCTOR0:def.39">FUNCTOR0:def 39</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font> as   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>;<br/><a NAME="E5:40_1"/>
( <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="functor1.html#T12" title="FUNCTOR1:th.12">FUNCTOR1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:40_1"/>
<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R2" title="FUNCTOR0:pred.2">are_anti-isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D40" title="FUNCTOR0:def.40">FUNCTOR0:def 40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:40"/>
<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R2" title="FUNCTOR0:pred.2">are_anti-isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">F</font> being    <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font><span class="kw"> such that </span><br/><a NAME="E6:40"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
( <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D40" title="FUNCTOR0:def.40">FUNCTOR0:def 40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font> as   <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/>
<a NAME="E8:40"/>
( <font color="Maroon" title="c5">F</font> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Maroon" title="c5">F</font> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> is  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="functor1.html#T12" title="FUNCTOR1:th.12">FUNCTOR1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:40"/>
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D39" title="FUNCTOR0:def.39">FUNCTOR0:def 39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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