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<div class="add">

<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="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<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="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" 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="E1:64"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green">E58</font></i>: 
( not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E3:64"/><i><font color="Green">E59</font></i>: 
<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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is  not   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is  not   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is  not   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is  not   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is  not   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:64_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>11</sub></font> &amp; <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="afproj.html#E1:64_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:64_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:64_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:64_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:64_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:64_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:64_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:64_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:64_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E12:64_1_1"/><i><font color="Green">E67</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>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="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E3:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green">Lemma43</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>;<br/><a NAME="E14:64_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E3:64_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E5:64_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E7:64_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E9:64_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E11:64_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_4.html#T27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</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="E16:64_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><b>consider </b><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="E18:64_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><b>set </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/><a NAME="E19:64_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/><a NAME="E20:64_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E22:64_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T38">AFF_4:38</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E24:64_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T38">AFF_4:38</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E26:64_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E20:64_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="aff_4.html#T39">AFF_4:39</a></i>;<br/><a NAME="E27:64_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E16:64_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E18:64_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="afproj.html#E22:64_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="afproj.html#E24:64_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_4.html#T28">AFF_4:28</a></i>;<br/><a NAME="E28:64_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E22:64_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E2:64_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E3:64_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E5:64_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E22:64_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="afproj.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:64_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E24:64_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E2:64_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E3:64_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E9:64_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E24:64_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afproj.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:64_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>27</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>27</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E7:64_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E11:64_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E22:64_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="afproj.html#E24:64_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="afproj.html#E28:64_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="afproj.html#E29:64_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E2:64_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>27</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T56">AFF_4:56</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E22:64_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="afproj.html#E24:64_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="aff_4.html#T43">AFF_4:43</a></i>;<br/>


</div><b>end;</b></div><a NAME="E31:64_1_1"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_1_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:64_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:64_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E19:64_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E4:64_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E5:64_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E3:64_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E7:64_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:64_1_1"/><i><font color="Green">E81</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_1_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:64_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:64_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E19:64_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E4:64_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E5:64_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E3:64_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E11:64_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><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="E34:64_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E19:64_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E22:64_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="afproj.html#E27:64_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="afproj.html#E30:64_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="afproj.html#E31:64_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="aff_4.html#T22">AFF_4:22</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>30</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="E36:64_1_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E19:64_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E24:64_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="afproj.html#E27:64_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="afproj.html#E30:64_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="afproj.html#E32:64_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="aff_4.html#T22">AFF_4:22</a></i>;<br/><b>set </b><font color="Maroon">c<sub>31</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>32</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font>;<br/><a NAME="E37:64_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_1_6"/>
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then </b><font color="Maroon">c<sub>2</sub></font> = 
 <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="afproj.html#E5:64_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="afproj.html#E9:64_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E3:64_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E3:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E38:64_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E16:64_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E18:64_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E39:64_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_1_7"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/>

<font color="Maroon">c<sub>2</sub></font> = 
 <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="afproj.html#E5:64_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>
<br/>.= 
 <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="aff_4.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E16:64_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E19:64_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E34:64_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="afproj.html#E1:64_1_1_7"><i><font color="Green">E87</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>
<br/>.= 
 <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E19:64_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="afproj.html#E3:64_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E3:64_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E40:64_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>30</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_1_8"/><i><font color="Green">E88</font></i>: 
not <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>30</sub></font>
 ;<br/>

<font color="Maroon">c<sub>3</sub></font> = 
 <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="afproj.html#E7:64_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>
<br/>.= 
 <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E34:64_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="afproj.html#E36:64_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="afproj.html#E39:64_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="afproj.html#E1:64_1_1_8"><i><font color="Green">E88</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="afproj.html#E11:64_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E3:64_1_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E3:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E41:64_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E38:64_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E42:64_1_1"/><i><font color="Green">E89</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>28</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E19:64_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E26:64_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="afproj.html#E34:64_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="afproj.html#E36:64_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="afproj.html#E38:64_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="afproj.html#E40:64_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="aff_4.html#T19">AFF_4:19</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E44:64_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T38">AFF_4:38</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E46:64_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="aff_4.html#T38">AFF_4:38</a></i>;<br/><a NAME="E47:64_1_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E48:64_1_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_9"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>36</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> as   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E44:64_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="afproj.html#E46:64_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E2:64_1_1_9"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>36</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E5:64_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E7:64_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E9:64_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E11:64_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E44:64_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="afproj.html#E46:64_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="afproj.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_9"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E12:64_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E56"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E49:64_1_1"/><i><font color="Green">E94</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E16:64_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E18:64_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="afproj.html#E34:64_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="afproj.html#E36:64_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="afproj.html#E38:64_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="afproj.html#E40:64_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="afproj.html#E44:64_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="afproj.html#E46:64_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="aff_4.html#T19">AFF_4:19</a></i>;<br/><div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1_10"/><i><font color="Green">E96</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>35</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span>)</span>, <font color="Maroon">c<sub>36</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E41:64_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>take </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font>;<br/><a NAME="E3:64_1_1_10"/>
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E41:64_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="afproj.html#E1:64_1_1_10"><i><font color="Green">E96</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1_10"/>
( <font color="Maroon">c<sub>37</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E41:64_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="afproj.html#E44:64_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="afproj.html#E46:64_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="afproj.html#E48:64_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="afproj.html#E49:64_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="afproj.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_11"><b>now </b></a><div class="add"><b>given </b><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><a NAME="E1:64_1_1_11"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font> )
 ;<br/><a NAME="E2:64_1_1_11"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>35</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_1_11_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/>

<a NAME="E2:64_1_1_11_1"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E16:64_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E18:64_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="afproj.html#E41:64_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="afproj.html#E47:64_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="afproj.html#E1:64_1_1_11"><i><font color="Green">E96</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E37:64_1_1"><i><font color="Green">E84</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>36</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>35</sub></font>;<br/><a NAME="E3:64_1_1_11"/><i><font color="Green">E98</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>35</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E2:64_1_1_11"><i><font color="Green">E97</font></i></a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>37</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>35</sub></font></span>)</span> as   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>take </b><font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>37</sub></font>;<br/><a NAME="E5:64_1_1_11"/>
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>35</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>35</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E14:64_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E44:64_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="afproj.html#E46:64_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="afproj.html#E49:64_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="afproj.html#E1:64_1_1_11"><i><font color="Green">E96</font></i></a>, <a class="txt" href="afproj.html#E2:64_1_1_11"><i><font color="Green">E97</font></i></a>, <a class="ref" href="aff_4.html#T19">AFF_4:19</a></i>;<br/><b>hence </b><a NAME="E6:64_1_1_11"/>
( <font color="Maroon">c<sub>38</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E44:64_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="afproj.html#E46:64_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="afproj.html#E48:64_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="afproj.html#E3:64_1_1_11"><i><font color="Green">E98</font></i></a>, <a class="ref" href="afproj.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E52:64_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E26:64_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="afproj.html#E41:64_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="afproj.html#E42:64_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="afproj.html#E50:64_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="aff_4.html#T22">AFF_4:22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:64_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E3:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="afproj.html#E1:64_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E68"><i><font color="Green">Lemma55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:64"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:64"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E2:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E3:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="afproj.html#E66"><i><font color="Green">Lemma53</font></i></a></i>;<br/>


</div>
