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<div class="add">

<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">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">c</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><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E29</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a'</font>
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E30</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>and </b><br/><a NAME="E3:22"/><i><font color="Green">E31</font></i>: 
<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>
 <b>and </b><br/><a NAME="E4:22"/><i><font color="Green">E32</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E5:22"/><i><font color="Green">E33</font></i>: 
 <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="E6:22"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green">E35</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green">E36</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</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></i>;<br/>
<a NAME="E9:22"/><i><font color="Green">E37</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">a'</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E10:22"/><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">c</font>
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/>
<a NAME="E11:22"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, , , <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E12:22"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> implies  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E2:22_1"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font>,<font color="Maroon">c</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font>,<font color="Maroon">c'</font> )
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E3:22_1"/><i><font color="Green">E43</font></i>: 
<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>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E4:22_1"/>
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</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/>
<b>hence </b><a NAME="E5:22_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> )
 ;<br/><a NAME="E2:22_2"/><i><font color="Green">E45</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_1"/><i><font color="Green">E46</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">a'</font>
 ;<br/>

<a NAME="E2:22_2_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">b'</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></i>;<br/>
<a NAME="E3:22_2_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E4:22_2_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="E3:22_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> implies  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 ;<br/>

<a NAME="E2:22_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">b'</font>,<font color="Maroon">b</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></i>;<br/>
<a NAME="E3:22_2_2"/><i><font color="Green">E52</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c'</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/>
<a NAME="E4:22_2_2"/>
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_2_1"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:22_2_2_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E3:22_2_2_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="E5:22_2_2"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">a'</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:22_2_2"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font>
 ;<br/><a NAME="E2:22_2_3"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E3:22_2_3"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E5:22_2_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c'</font>,<font color="Maroon">a</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">x</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">x</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#T32">DIRAF:32</a></i>;<br/><a NAME="E6:22_2_3"/>
<font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/><a NAME="E7:22_2_3"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">c'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, , , <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/><a NAME="E8:22_2_3"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c'</font>,<font color="Maroon">x</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E9:22_2_3"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">a</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E10:22_2_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:22_2_3_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c'</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</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="E3:22_2_3_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>
<a NAME="E4:22_2_3_1"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T40">DIRAF:40</a></i>;<br/>
<a NAME="E5:22_2_3_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E6:22_2_3_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E7:22_2_3_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</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/>
<b>hence </b><a NAME="E8:22_2_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:22_2_3"/><i><font color="Green">E62</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E12:22_2_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_2_3_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E2:22_2_3_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">x</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#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E3:22_2_3_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>
 
<b>by </b><i>, , <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<b>hence </b><a NAME="E4:22_2_3_2"/>
<font color="Maroon">x</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:22_2_3"/>
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_3_3"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:22_2_3_3"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">b</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="E3:22_2_3_3"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">b'</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#T38">DIRAF:38</a></i>;<br/>
<b>hence </b><a NAME="E4:22_2_3_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:22_2_3"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="pasch.html#T7" target="_self">Th7</a>, <a class="txt" href="pasch.html#E1:22"><i><font color="Green">E29</font></i></a>, </i>;<br/><a NAME="E15:22_2_3"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c'</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E16:22_2_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">b'</font>,<font color="Maroon">x</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:22_2_3_4"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E2:22_2_3_4"/><b>then </b>
<font color="Maroon">b'</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b'</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E3:22_2_3_4"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b'</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<b>hence </b><a NAME="E4:22_2_3_4"/>
<font color="Maroon">b'</font>,<font color="Maroon">x</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/>


</div><b>end;</b></div><a NAME="E17:22_2_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_3_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E2:22_2_3_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</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="E3:22_2_3_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp; <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>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>
<a NAME="E4:22_2_3_5"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E5:22_2_3_5"/>
contradiction
 <b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:22_2_3"/>
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_3_6"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:22_2_3_6"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">x</font>,<font color="Maroon">c'</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">x</font>,<font color="Maroon">b'</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">x</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="E3:22_2_3_6"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:22"><i><font color="Green">E32</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>
<b>hence </b><a NAME="E4:22_2_3_6"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:22_2_3"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, <a class="txt" href="pasch.html#E2:22"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pasch.html#E3:22"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>hence </b><a NAME="E20:22_2_3"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:22_2"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:22"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/>


</div>
