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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="anproj_2.html#V1" target="_self">up-3-dimensional</a> <a href="rlvect_1.html#NM2">RealLinearSpace</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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D13" target="_self">ANPROJ_2:def 13</a><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E79</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>5</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>5</sub></font> &amp; <font color="Maroon">c<sub>3</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>3</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>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>7</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>6</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>6</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="E2:88"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:88"/><i><font color="Green">E81</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="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E4:88"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>consider </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> 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="E6:88"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <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="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green">E81</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</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="E8:88"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>17</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green">E81</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E9:88"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<a NAME="E1:88_1"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E8:88"><i><font color="Green">E84</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_1"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8:88"><i><font color="Green">E84</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</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="E11:88"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>20</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green">E81</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E12:88"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>proof </b></a><div class="add">

<a NAME="E1:88_2"/>
 <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_2"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</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="E14:88"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>22</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>23</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <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="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green">E81</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E15:88"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3"><b>proof </b></a><div class="add">

<a NAME="E1:88_3"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_3"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E14:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:88"/>
not <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:88"><i><font color="Green">E80</font></i></a>, <a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E17:88"/><b>then </b><i><font color="Green">E90</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>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="anproj_2.html#R4" target="_self">lie_on_an_angle</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E9:88"><i><font color="Green">E85</font></i></a>, <a class="txt" href="anproj_2.html#E12:88"><i><font color="Green">E87</font></i></a>, <a class="txt" href="anproj_2.html#E15:88"><i><font color="Green">E89</font></i></a>, <a class="ref" href="anproj_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E18:88"/>
( not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>20</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>23</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>23</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; not  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:88"><i><font color="Green">E79</font></i></a>, <a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E8:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E19:88"/><b>then </b><i><font color="Green">E91</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>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="anproj_2.html#R5" target="_self">are_half_mutually_not_Prop</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E20:88"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E8:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</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="E22:88"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>25</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>26</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E82</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E23:88"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_4"><b>proof </b></a><div class="add">

<a NAME="E1:88_4"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E22:88"><i><font color="Green">E93</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_4"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E22:88"><i><font color="Green">E93</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</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="E25:88"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>28</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>29</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E82</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E26:88"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_5"><b>proof </b></a><div class="add">

<a NAME="E1:88_5"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>26</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E22:88"><i><font color="Green">E93</font></i></a>, <a class="txt" href="anproj_2.html#E25:88"><i><font color="Green">E95</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_5"/>
<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E25:88"><i><font color="Green">E95</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>31</sub></font>, <font color="Maroon">c<sub>32</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="E28:88"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>31</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>32</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>32</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E82</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E29:88"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>32</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_6"><b>proof </b></a><div class="add">

<a NAME="E1:88_6"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green">E88</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E97</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_6"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>32</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E97</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</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="E31:88"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>34</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>35</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E82</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E32:88"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>32</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_7"><b>proof </b></a><div class="add">

<a NAME="E1:88_7"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>32</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E97</font></i></a>, <a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E99</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_7"/>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>32</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green">E99</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>36</sub></font>, <font color="Maroon">c<sub>37</sub></font>, <font color="Maroon">c<sub>38</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="E34:88"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>37</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>38</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E82</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E35:88"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>38</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_8"><b>proof </b></a><div class="add">

<a NAME="E1:88_8"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green">E88</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green">E101</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_8"/>
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>38</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E34:88"><i><font color="Green">E101</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font>, <font color="Maroon">c<sub>41</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="E37:88"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>39</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K3">Dir</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>39</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>40</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>41</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>40</sub></font>,<font color="Maroon">c<sub>41</sub></font> <a href="anproj_1.html#R2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green">E82</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E38:88"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>38</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_9"><b>proof </b></a><div class="add">

<a NAME="E1:88_9"/>
(  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>40</sub></font>,<font color="Maroon">c<sub>20</sub></font> &amp;  <a href="anproj_1.html#R1">are_Prop</a> <font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>38</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:88"><i><font color="Green">E84</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green">E86</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green">E101</font></i></a>, <a class="txt" href="anproj_2.html#E37:88"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_1.html#T35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_9"/>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>38</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E37:88"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_1.html#T9">ANPROJ_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:88"/>
<font color="Maroon">c<sub>38</sub></font>,<font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="anproj_2.html#R1" target="_self">are_Prop_Vect</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E22:88"><i><font color="Green">E93</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E97</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green">E101</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E40:88"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font>,<font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="anproj_1.html#R2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green">E83</font></i></a>, <a class="txt" href="anproj_2.html#E17:88"><i><font color="Green">E90</font></i></a>, <a class="txt" href="anproj_2.html#E19:88"><i><font color="Green">E91</font></i></a>, <a class="txt" href="anproj_2.html#E20:88"><i><font color="Green">E92</font></i></a>, <a class="txt" href="anproj_2.html#E23:88"><i><font color="Green">E94</font></i></a>, <a class="txt" href="anproj_2.html#E26:88"><i><font color="Green">E96</font></i></a>, <a class="txt" href="anproj_2.html#E29:88"><i><font color="Green">E98</font></i></a>, <a class="txt" href="anproj_2.html#E32:88"><i><font color="Green">E100</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E38:88"><i><font color="Green">E104</font></i></a>, <a class="ref" href="anproj_2.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E41:88"/>
<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="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E22:88"><i><font color="Green">E93</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green">E97</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green">E101</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self">Th24</a></i>;<br/>


</div>
