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<b>let </b><font color="Maroon">OAS</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/>











<b>assume </b><a NAME="E1:39"/><i><font color="Green">E29</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">c</font> &amp; not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">d</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E2:39_1"/>
<font color="Maroon">d</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="diraf.html#D3">DIRAF:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">e1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">e1</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">e1</font> )
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E5:39_1"/><i><font color="Green">E34</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>,<font color="Maroon">e1</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E6:39_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">e1</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E7:39_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">e1</font>,<font color="Maroon">x</font>,<font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="diraf.html#T16">DIRAF:16</a></i>;<br/><a NAME="E8:39_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>,<font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E9:39_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 ;<br/>

<a NAME="E2:39_1_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:39_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font>
 <b>by </b><i>, , <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/><a NAME="E11:39_1"/><i><font color="Green">E39</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">e1</font>
 ;<br/>

<a NAME="E2:39_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">e1</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E3:39_1_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>,<font color="Maroon">e1</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E4:39_1_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">e1</font>,<font color="Maroon">x</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>,<font color="Maroon">e1</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E5:39_1_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">e1</font>,<font color="Maroon">x</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">e1</font>,<font color="Maroon">e1</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">e1</font>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<a NAME="E6:39_1_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">e1</font>,<font color="Maroon">b</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">e1</font>,<font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T41">DIRAF:41</a></i>;<br/>
<b>hence </b><a NAME="E7:39_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="pasch.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E13:39_1"/><i><font color="Green">E41</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">e1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E14:39_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="pasch.html#T7" target="_self">Th7</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E15:39_1"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="pasch.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E16:39_1"/><b>then </b>
<font color="Maroon">d</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E17:39_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">d</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E18:39_1"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><b>hence </b><a NAME="E19:39_1"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Olive">y</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Olive">y</font>,<font color="Maroon">x</font>,<font color="Maroon">d</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E2:39_2"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">c</font>;<br/><b>thus </b><a NAME="E2:39_3"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">d</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/><b>thus </b><a NAME="E3:39_3"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="diraf.html#T13">DIRAF:13</a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Olive">y</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Olive">y</font>,<font color="Maroon">x</font>,<font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/>


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