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<span class="kw">let </span><font color="Maroon" title="c1">AFV</font> be   <a href="afvect0.html#NM1" title="AFVECT0:NM.1">WeakAffVect</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> holds <br/> (  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Olive" title="b3">q</font>,<font color="Olive" title="b2">a</font>) iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">q</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">a</font>) iff ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">q</font> ) )</span><br/>







<div><span class="lab"><font color="Green" title="E28">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">a</font>)</span><br/><a NAME="E2:42_1"/>
(  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>) &amp;  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#D4" target="_self" title="AFVECT0:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_1"/>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="afvect0.html#T24" target="_self" title="AFVECT0:th.24">Th29</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">a</font>)
 ; <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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">q</font> )</span><br/><a NAME="E2:42_2"/>
(  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>) &amp;  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#D4" target="_self" title="AFVECT0:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_2"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="afvect0.html#T20" target="_self" title="AFVECT0:th.20">Th25</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">hence </span><a NAME="E3:42"/>
(  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">a</font>) iff ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">q</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E28">A1</font></span></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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