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<b>let </b><font color="Maroon" title="c1">A</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"> for <font color="Olive" title="b1">B</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b1">B</font>,<font color="Olive" title="b1">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a>  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span><br/><b>let </b><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_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font>; <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="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a>  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span><br/>



<b>set </b><font color="Maroon" title="c3">F</font> =  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span>;<br/>
<a NAME="E1:64"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  &amp; <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  )
 
<b>by </b><i><a class="ref" href="yellow18.html#D4" title="YELLOW18:def.4">YELLOW18:def 4</a></i>;<br/>
<b>thus </b><a NAME="E2:64"/>
( <font color="Maroon" title="c2">B</font> is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  )
 <b>by </b><i><a class="ref" href="yellow20.html#T49" target="_self" title="YELLOW20:th.49">Th49</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D5" target="_self" title="YELLOW20:def.5">YELLOW20:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being <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="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  st <br/>( <font color="Olive" title="b1">G</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> &amp; ( for <font color="Olive" title="b2">a'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> holds <br/><font color="Olive" title="b1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b3">a</font> ) &amp; ( for <font color="Olive" title="b2">b'</font>, <font color="Olive" title="b3">c'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b'</font>,<font color="Olive" title="b3">c'</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>  &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b3">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">c</font> holds <br/>for <font color="Olive" title="b6">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b'</font>,<font color="Olive" title="b3">c'</font><br/> for <font color="Olive" title="b7">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>  st <font color="Olive" title="b6">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f</font> holds <br/><font color="Olive" title="b1">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b6">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">f</font> ) )</span><br/>

<b>take </b><font color="Maroon" title="c4">G</font> =  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">G</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> &amp; ( for <font color="Olive" title="b1">a'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font> ) &amp; ( for <font color="Olive" title="b1">b'</font>, <font color="Olive" title="b2">c'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</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>  &amp; <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font> ) )</span><br/>

<b>thus </b><a NAME="E3:64"/>
<font color="Maroon" title="c4">G</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font> ) &amp; ( for <font color="Olive" title="b1">b'</font>, <font color="Olive" title="b2">c'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</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>  &amp; <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font> ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b'</font>, <font color="Olive" title="b2">c'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</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>  &amp; <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a1</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a1</font></span><br/><b>let </b><font color="Maroon" title="c6">a1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font> implies <font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a1</font> )</span><br/><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a1</font></span><br/><b>hence </b><font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
<font color="Maroon" title="c6">a1</font>
<b>by </b><i><a class="txt" href="#E1:64"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="yellow18.html#D5" title="YELLOW18:def.5">YELLOW18:def 5</a></i>
<br/>.= 
<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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a1</font>
<b>by </b><i><a class="txt" href="#E1:64"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="yellow18.html#D5" title="YELLOW18:def.5">YELLOW18:def 5</a></i>
;<br/> <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>let </b><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</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>  &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> holds <br/>for <font color="Olive" title="b3">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font><br/> for <font color="Olive" title="b4">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font>  st <font color="Olive" title="b3">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b3">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">f</font></span><br/><b>let </b><font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c8">c1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</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>  &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c1</font> implies for <font color="Olive" title="b1">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font>  st <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font> )</span><br/>



<b>assume </b><a NAME="E5:64"/><i><font color="Green" title="E37">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">b</font>,<font color="Maroon" title="c6">c</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>  &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font>  st <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font></span><br/>

<b>let </b><font color="Maroon" title="c9">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font>  st <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c10">f1</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> implies <font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f1</font> )</span><br/>



<b>assume </b><a NAME="E6:64"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f1</font></span><br/>

<a NAME="E7:64"/><i><font color="Green" title="E39">A4</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">b</font>,<font color="Maroon" title="c6">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="#E5:64"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green" title="E40">A5</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"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">c1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">b1</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="ref" href="functor0.html#D20" title="FUNCTOR0:def.20">FUNCTOR0:def 20</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c9">f</font> = 
<font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="#E1:64"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E5:64"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="yellow18.html#D5" title="YELLOW18:def.5">YELLOW18:def 5</a></i>
<br/>.= 
<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>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c10">f1</font>
<b>by </b><i><a class="txt" href="#E1:64"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E6:64"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E7:64"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="yellow18.html#D5" title="YELLOW18:def.5">YELLOW18:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K7" title="FUNCTOR0:func.7">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f1</font>
<b>by </b><i><a class="txt" href="#E7:64"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="#E8:64"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="functor0.html#D17" title="FUNCTOR0:def.17">FUNCTOR0:def 17</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>
