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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">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<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">c</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>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">b</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="E2:13"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E33</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>
 ;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green">E34</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">d</font>
 <b>and </b><br/><a NAME="E4:13_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:13_1_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">c</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#T5">DIRAF:5</a></i>;<br/><a NAME="E3:13_1_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">p</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#T6">DIRAF:6</a></i>;<br/><a NAME="E4:13_1_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/><a NAME="E5:13_1_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</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="E6:13_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:13_1_2"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E3:13_1_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E4:13_1_2"/><b>then </b>
<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>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E5:13_1_2"/><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#D4">DIRAF:def 4</a></i>;<br/><a NAME="E6:13_1_2"/><b>then </b>
 <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/><b>hence </b><a NAME="E7:13_1_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="E7:13_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E8:13_1"/>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Olive">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Olive">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>
 ;<br/><a NAME="E2:13_2"/><b>then </b>
 <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#T36">DIRAF:36</a></i>;<br/><a NAME="E3:13_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<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#D5">DIRAF:def 5</a></i>;<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_1"/>
<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>
 ;<br/><a NAME="E2:13_2_1"/><b>then </b><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">b</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E3:13_2_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</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="E4:13_2_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</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/><b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E6:13_2_1"/><i><font color="Green">E42</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E7:13_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="diraf.html#T17">DIRAF:17</a></i>;<br/><a NAME="E8:13_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/><a NAME="E9:13_2_1"/>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E10:13_2_1"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E11:13_2_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E12:13_2_1"/><b>then </b><i><font color="Green">E45</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">d</font>
 <b>by </b><i>, , <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E13:13_2_1"/>
<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#T6">DIRAF:6</a></i>;<br/><a NAME="E14:13_2_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="E15:13_2_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="E16:13_2_1"/><b>then </b>
(  <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">b</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E17:13_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</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="E18:13_2_1"/>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">c</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">d</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E19:13_2_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="diraf.html#T40">DIRAF:40</a></i>;<br/><a NAME="E20:13_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/><b>hence </b><a NAME="E21:13_2_1"/>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Olive">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Olive">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2_2"/>
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font>
 ;<br/><a NAME="E2:13_2_2"/><b>then </b><i><font color="Green">E47</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="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E3:13_2_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E4:13_2_2"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</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="E5:13_2_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">b</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="E6:13_2_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">b</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">c</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="E7:13_2_2"/><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">b</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><a NAME="E8:13_2_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T40">DIRAF:40</a></i>;<br/><b>hence </b><a NAME="E9:13_2_2"/>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Olive">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Olive">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:13_2"/>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Olive">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Olive">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</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="E6:13"/>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Olive">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c</font>,<font color="Olive">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> )
 <b>by </b><i/>;<br/>


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