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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>and </b><br/><a NAME="E4:22"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Q</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/>

<a NAME="E5:22"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E6:22"/><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">b'</font>
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E7:22"/>
 <a href="aff_1.html#R1">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="aff_1.html#T33">AFF_1:33</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">AS</font><b> such that </b><br/><a NAME="E9:22"/><i><font color="Green">E45</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a'</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E10:22"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E11:22"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E12:22"/><b>then </b><i><font color="Green">E57</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="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/>
<a NAME="E13:22"/><i><font color="Green">E59</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E14:22"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:22_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E3:22_1"/>
 <a href="aff_1.html#R1">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="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E5:22_1"/><i><font color="Green">E62</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E6:22_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:22_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E8:22_1"/>
<font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E9:22_1"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E10:22_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">a</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E11:22_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E12:22_1"/>
 ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:22"/>
( <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Q</font> or  ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> ) )
 <b>by </b><i>, , , , , , <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>


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