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<b>let </b><font color="Maroon">AFV</font> be   <a href="afvect0.html#NM1">WeakAffVect</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font>;<br/>





<div><i><font color="Green">E26</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">E28</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/><div><i><font color="Green">E29</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">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">q</font> = <font color="Maroon">a</font>;<br/><a NAME="E2:3_1_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="afvect0.html#T3">AFVECT0:3</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="diraf.html#D4">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">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font><b> such that </b><br/><a NAME="E3:3_1_2"/><i><font color="Green">E32</font></i>: 
 <a href="afvect0.html#R2">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="afvect0.html#T24">AFVECT0:24</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font><b> such that </b><br/><a NAME="E5:3_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="afvect0.html#D1">AFVECT0:def 1</a></i>;<br/><a NAME="E6:3_1_2"/>
 <a href="afvect0.html#R1">MDist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="afvect0.html#D2">AFVECT0:def 2</a></i>;<br/><a NAME="E7:3_1_2"/><b>then </b>
 <a href="afvect0.html#R1">MDist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="afvect0.html#T19">AFVECT0:19</a></i>;<br/><a NAME="E8:3_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="afvect0.html#R2">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="afvect0.html#T28">AFVECT0:28</a></i>;<br/><a NAME="E9:3_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="afvect0.html#D3">AFVECT0:def 3</a></i>;<br/><a NAME="E10:3_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="afvect0.html#D3">AFVECT0:def 3</a></i>;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">p</font>;<br/><b>take </b><font color="Maroon">q</font> = <font color="Maroon">q</font>;<br/><b>thus </b><a NAME="E11:3_1_2"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, , , <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:3_1"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="diraf.html#R2">'||'</a> <font color="Olive">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Olive">q</font> <a href="diraf.html#R2">'||'</a> <font color="Olive">q</font>,<font color="Maroon">b</font> )
 <b>by </b><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">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font><b> such that </b><a NAME="E1:3_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">q</font>,<font color="Maroon">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">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:3_2_1"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:3_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="afvect0.html#R2">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font> &amp;  <a href="afvect0.html#R2">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="afvect0.html#D3">AFVECT0:def 3</a></i>;<br/><div><i><font color="Green">E40</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">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:3_2_1_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E3:3_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="afvect0.html#D1">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">E41</font></i>: 
 <a href="afvect0.html#R1">MDist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 ;<br/><a NAME="E2:3_2_1_2"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> or <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/><a NAME="E3:3_2_1_2"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> or <font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="afvect0.html#T4">AFVECT0:4</a></i>;<br/><a NAME="E4:3_2_1_2"/><b>then </b>
 <a href="afvect0.html#R1">MDist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="afvect0.html#T19">AFVECT0:19</a></i>;<br/><b>hence </b><a NAME="E5:3_2_1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="afvect0.html#D2">AFVECT0:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:3_2_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="afvect0.html#T25">AFVECT0:25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:3_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="afvect0.html#T3">AFVECT0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> iff  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFV</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="diraf.html#R2">'||'</a> <font color="Olive">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Olive">q</font> <a href="diraf.html#R2">'||'</a> <font color="Olive">q</font>,<font color="Maroon">b</font> ) )
 <b>by </b><i/>;<br/>


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