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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:41"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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="E2:41"/><b>then </b><i><font color="Green" title="E20">A1</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="yellow18.html#T16" target="_self" title="YELLOW18:th.16">Th16</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>assume </b><a NAME="E1:41_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:()"><i><font color="Red">::  thesis: </font></i></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/><b>then consider </b><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><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <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> )
 <b>by </b><i><a class="ref" href="functor0.html#D40" title="FUNCTOR0:def.40">FUNCTOR0:def 40</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="yellow18.html#E3:41_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/><a NAME="E5:41_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> )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:41"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="yellow18.html#E3:41_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="functor1.html#T13" title="FUNCTOR1:th.13">FUNCTOR1:13</a></i>;<br/><b>hence </b><a NAME="E6:41_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> 
 <b>by </b><i><a class="ref" href="functor0.html#D41" title="FUNCTOR0:def.41">FUNCTOR0:def 41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:41"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green" title="E21">A3</font></i>: 
( <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> )
 <b>by </b><i><a class="ref" href="functor0.html#D41" title="FUNCTOR0:def.41">FUNCTOR0:def 41</a></i>;<br/>
<b>reconsider </b><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> <b>by </b><i><a class="txt" href="yellow18.html#E6:41"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>
<a NAME="E8:41"/>
( <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> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:41"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="yellow18.html#E6:41"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="functor1.html#T13" title="FUNCTOR1:th.13">FUNCTOR1:13</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
<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> 
 <b>by </b><i><a class="ref" href="functor0.html#D40" title="FUNCTOR0:def.40">FUNCTOR0:def 40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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