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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">p</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#R1" title="AFVECT0:pred.1">MDist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> iff  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Olive" title="b3">p</font>,<font color="Olive" title="b1">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Olive" title="b3">p</font>,<font color="Olive" title="b2">b</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">p</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#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> iff  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>) )</span><br/>







<div><span class="lab"><font color="Green" title="E26">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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#R1" title="AFVECT0:pred.1">MDist</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>)</span><br/><a NAME="E2:40_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#D2" target="_self" title="AFVECT0:def.2">Def2</a></span>;<br/><a NAME="E3:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>) <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T33" target="_self" title="AFVECT0:th.33">Th40</a></span>;<br/><a NAME="E4:40_1"/>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="afvect0.html#T30" target="_self" title="AFVECT0:th.30">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E5:40_1"/>
 <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="afvect0.html#D2" target="_self" title="AFVECT0:def.2">Def2</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="40_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</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#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><br/><a NAME="E2:40_2"/><span class="kw">then </span>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>) <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#D2" target="_self" title="AFVECT0:def.2">Def2</a></span>;<br/><a NAME="E3:40_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T33" target="_self" title="AFVECT0:th.33">Th40</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_2"/>
 <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="afvect0.html#D2" target="_self" title="AFVECT0:def.2">Def2</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="E3:40"/>
(  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> iff  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font>), <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E26">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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