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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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<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">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">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green">E29</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font>
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b'</font>
 <b>and </b><br/><a NAME="E4:25"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>
 ;<br/>

<a NAME="E5:25"/><i><font color="Green">E33</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a'</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E6:25"/>
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E7:25"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E8:25"/>
<font color="Maroon">p</font>,<font color="Maroon">a'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E9:25"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E10:25"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E12:25"/><i><font color="Green">E37</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T17">DIRAF:17</a></i>;<br/>
<a NAME="E13:25"/><i><font color="Green">E38</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">c</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E14:25"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E16:25"/><i><font color="Green">E40</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">c'</font>
 <b>and </b><br/><a NAME="E17:25"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E18:25"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:25_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">a'</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="E5:25_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">a'</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/>
<b>hence </b><a NAME="E6:25_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:25"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:25_2"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</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="E3:25_2"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E4:25_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E5:25_2"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">a'</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:25_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:25"/>
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<a NAME="E21:25"/><b>then </b>
<font color="Maroon">c</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="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">b''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E23:25"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c'</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b''</font>
 <b>and </b><br/><a NAME="E24:25"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c'</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">b''</font>
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E25:25"/>
<font color="Maroon">a'</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E26:25"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">a'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">c'</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b''</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E27:25"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">a'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">c'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b''</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E28:25"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">a'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p</font>,<font color="Maroon">b''</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E29:25"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a'</font>,<font color="Maroon">b''</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E30:25"/>
( <font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c'</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c'</font>,<font color="Maroon">c</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">b''</font> )
 
<b>by </b><i>, , <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E31:25"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">b''</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E32:25"/><b>then </b>
( <font color="Maroon">c'</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">c'</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i>, , , , , , , , <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<b>hence </b><a NAME="E33:25"/>
<font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>


</div>
