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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="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  iff <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R3" title="YELLOW18:pred.3">are_dual</a>  ) )</span><br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E26">A1</font></i>: 
<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="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  iff <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R3" title="YELLOW18:pred.3">are_dual</a>  )</span><br/>

<a NAME="E2:53"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R3" title="YELLOW18:pred.3">are_dual</a>  iff <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  )
 
<b>by </b><i><a class="ref" href="yellow18.html#D6" target="_self" title="YELLOW18:def.6">Def6</a></i>;<br/>
<a NAME="E3:53"/>
<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c3">C</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" target="_self" title="YELLOW18:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:53"/>
( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  iff <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R3" title="YELLOW18:pred.3">are_dual</a>  )
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E2:53"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow18.html#T26" target="_self" title="YELLOW18:th.26">Th26</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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