<?xml version="1.0"?>
<div class="add">

<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#R3" title="YELLOW18:pred.3">are_dual</a>  &amp; <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a>  implies <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R3" title="YELLOW18:pred.3">are_dual</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a> 
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow18.html#D6" target="_self" onmouseover="rs('yellow18/D6')" onmouseout="rh()">YELLOW18:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="yellow18.html#R3" title="YELLOW18:pred.3">are_dual</a> </span><br/>

<a NAME="E3:51"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow18.html#D4" target="_self" onmouseover="rs('yellow18/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:51"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow18.html#D4" target="_self" onmouseover="rs('yellow18/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E5:51"/><span class="kw">then </span>
<font color="Maroon" title="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> ,<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="yellow18.html#T25" target="_self" onmouseover="rs('yellow18/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51"/>
<font color="Maroon" title="c1">A</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow18.html#T4" target="_self" onmouseover="rs('yellow18/T4')" onmouseout="rh()">Th4</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow18.html#D6" target="_self" onmouseover="rs('yellow18/D6')" onmouseout="rh()">YELLOW18:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
