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<b>let </b><font color="Maroon" title="c1">AFV</font> be   <a href="afvect0.html#NM1" title="AFVECT0:NM.1">WeakAffVect</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font>;<br/>





<div><i><font color="Green" title="E3">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1"/><i><font color="Green" title="E4">A2</font></i>: 
<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>
 ;<br/><div><i><font color="Green" title="E5">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><b>take </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">a</font>;<br/><b>take </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c2">a</font>;<br/><a NAME="E2:3_1_1"/>
( <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="c4">p</font>,<font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="afvect01.html#E1:3_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="afvect0.html#T3" title="AFVECT0:th.3">AFVECT0:3</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_2"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/><b>consider </b><font color="Maroon" title="c4">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font><b> such that </b><br/><a NAME="E3:3_1_2"/><i><font color="Green" title="E7">A6</font></i>: 
 <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="afvect0.html#T24" title="AFVECT0:th.24">AFVECT0:24</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font><b> such that </b><br/><a NAME="E5:3_1_2"/><i><font color="Green" title="E8">A7</font></i>: 
<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="c4">p</font>,<font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="ref" href="afvect0.html#D1" title="AFVECT0:def.1">AFVECT0:def 1</a></i>;<br/><a NAME="E6:3_1_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>
 <b>by </b><i><a class="txt" href="afvect01.html#E1:3_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="afvect01.html#E1:3_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="afvect0.html#D2" title="AFVECT0:def.2">AFVECT0:def 2</a></i>;<br/><a NAME="E7:3_1_2"/><b>then </b><i><font color="Green" title="E9">A8</font></i>: 
 <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="afvect01.html#E3:3_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="afvect01.html#E5:3_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="afvect0.html#T19" title="AFVECT0:th.19">AFVECT0:19</a>, <a class="ref" href="afvect0.html#T28" title="AFVECT0:th.28">AFVECT0:28</a></i>;<br/><a NAME="E8:3_1_2"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="afvect01.html#E3:3_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="afvect0.html#D3" title="AFVECT0:def.3">AFVECT0:def 3</a></i>;<br/><a NAME="E9:3_1_2"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="afvect01.html#E7:3_1_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="afvect0.html#D3" title="AFVECT0:def.3">AFVECT0:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">p</font>;<br/><b>take </b><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c5">q</font>;<br/><b>thus </b><a NAME="E10:3_1_2"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">q</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="afvect01.html#E5:3_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="afvect01.html#E8:3_1_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="afvect01.html#E9:3_1_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:3_1"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">p</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">q</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="afvect01.html#E2:3_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font><b> such that </b><a NAME="E2:3_2"/><i><font color="Green" title="E4">A11</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_2_1"/><i><font color="Green" title="E5">A12</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E2:3_2_1"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="afvect01.html#E2:3_2"><i><font color="Green" title="E4">A11</font></i></a>, <a class="txt" href="afvect01.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E3:3_2_1"/><b>then </b><i><font color="Green" title="E6">A13</font></i>: 
(  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font> &amp;  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="afvect0.html#D3" title="AFVECT0:def.3">AFVECT0:def 3</a></i>;<br/><div><i><font color="Green" title="E7">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_2_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 ;<br/><a NAME="E2:3_2_1_1"/><b>then </b>
<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="c4">p</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="afvect01.html#E2:3_2"><i><font color="Green" title="E4">A11</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E3:3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afvect01.html#E1:3_2_1"><i><font color="Green" title="E5">A12</font></i></a>, <a class="ref" href="afvect0.html#D1" title="AFVECT0:def.1">AFVECT0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_2_1_2"/><i><font color="Green" title="E8">A15</font></i>: 
 <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>
 ;<br/><a NAME="E2:3_2_1_2"/>
( <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="c4">p</font>,<font color="Maroon" title="c5">q</font> or <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="c5">q</font>,<font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="afvect01.html#E2:3_2"><i><font color="Green" title="E4">A11</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><a NAME="E3:3_2_1_2"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="afvect01.html#E1:3_2_1_2"><i><font color="Green" title="E8">A15</font></i></a>, <a class="ref" href="afvect0.html#T4" title="AFVECT0:th.4">AFVECT0:4</a>, <a class="ref" href="afvect0.html#T19" title="AFVECT0:th.19">AFVECT0:19</a></i>;<br/><b>hence </b><a NAME="E4:3_2_1_2"/>
<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>
 <b>by </b><i><a class="ref" href="afvect0.html#D2" title="AFVECT0:def.2">AFVECT0:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:3_2_1"/>
<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>
 <b>by </b><i><a class="txt" href="afvect01.html#E3:3_2_1"><i><font color="Green" title="E6">A13</font></i></a>, <a class="txt" href="afvect01.html#E4:3_2_1"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="afvect0.html#T25" title="AFVECT0:th.25">AFVECT0:25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:3_2"/>
<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>
 <b>by </b><i><a class="ref" href="afvect0.html#T3" title="AFVECT0:th.3">AFVECT0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
( <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> iff  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">p</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">q</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">b</font> ) )
 <b>by </b><i><a class="txt" href="afvect01.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>


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