<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="anproj_2.html#NM1">CollProjectiveSpace</a>;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font>, <font color="Olive">b<sub>10</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>7</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font>,<font color="Olive">b<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</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>, <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>, <font color="Maroon">c<sub>11</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:23"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="incproj.html#R1" target="_self">are_mutually_different</a> 
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:23"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="incproj.html#E2:23"><i><font color="Green">E24</font></i></a>, <a class="ref" href="incproj.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>30</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>30</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="incproj.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E8:23"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="incproj.html#E6:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="incproj.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E9:23"/>
( not <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1"/>
<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E4:23_1_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="incproj.html#E4:23"><i><font color="Green">E26</font></i></a>, <a class="txt" href="incproj.html#E6:23"><i><font color="Green">E27</font></i></a>, <a class="txt" href="incproj.html#E3:23_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E5:23_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E3:23"><i><font color="Green">E25</font></i></a>, <a class="ref" href="incproj.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2"/>
<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:23_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E4:23_1_2"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="incproj.html#E4:23"><i><font color="Green">E26</font></i></a>, <a class="txt" href="incproj.html#E6:23"><i><font color="Green">E27</font></i></a>, <a class="txt" href="incproj.html#E3:23_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E5:23_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E3:23"><i><font color="Green">E25</font></i></a>, <a class="ref" href="incproj.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_3"/>
<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:23_1_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E4:23_1_3"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="incproj.html#E4:23"><i><font color="Green">E26</font></i></a>, <a class="txt" href="incproj.html#E6:23"><i><font color="Green">E27</font></i></a>, <a class="txt" href="incproj.html#E3:23_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="incproj.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E5:23_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E3:23"><i><font color="Green">E25</font></i></a>, <a class="ref" href="incproj.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:23_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incproj.html#E1:23_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="incproj.html#E2:23_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="incproj.html#E3:23_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:23"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="incproj.html#E1:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="incproj.html#E4:23"><i><font color="Green">E26</font></i></a>, <a class="txt" href="incproj.html#E7:23"><i><font color="Green">E28</font></i></a>, <a class="txt" href="incproj.html#E8:23"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E12:23"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E13:23"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="txt" href="incproj.html#E12:23"><i><font color="Green">E30</font></i></a>, <a class="ref" href="incproj.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E14:23"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="incproj.html#R4" target="_self">on</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div>
