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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E18</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E4:22"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/>

<a NAME="E5:22"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="aff_4.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="aff_4.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:22"/><i><font color="Green">E23</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E10:22"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E11:22"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="aff_4.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E9:22"><i><font color="Green">E23</font></i></a>, <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">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="aff_4.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/>
<a NAME="E13:22"/><i><font color="Green">E27</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i><a class="txt" href="aff_4.html#E12:22"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E14:22"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:22_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E6:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E13:22"><i><font color="Green">E27</font></i></a>, <a class="txt" href="aff_4.html#E14:22"><i><font color="Green">E28</font></i></a>, <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">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E11:22"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:22_1"/><i><font color="Green">E30</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E6:22_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:22_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:22_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E6:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E5:22_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E8:22_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E10:22"><i><font color="Green">E24</font></i></a>, <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">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E12:22"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_4.html#E6:22_1"><i><font color="Green">E31</font></i></a>, <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">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E5:22"><i><font color="Green">E21</font></i></a>, <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">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E7:22_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:22"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E5:22"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_4.html#E10:22"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_4.html#E12:22"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_4.html#E14:22"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>


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