<?xml version="1.0"?>
<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="E1:61"/><i><font color="Green">E54</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> )
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E55</font></i>: 
( <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> )
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green">E56</font></i>: 
( <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="E4:61"/><i><font color="Green">E57</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="E5:61"/><i><font color="Green">E58</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>
 <b>and </b><br/><a NAME="E6:61"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E7:61"/><i><font color="Green">E60</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#E1:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="afproj.html#E4:61"><i><font color="Green">E57</font></i></a>, <a class="txt" href="afproj.html#E5:61"><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/>
<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="E9:61"/><i><font color="Green">E61</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"><font color="Maroon">c<sub>11</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <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>11</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self">Th23</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="E11:61"/><i><font color="Green">E62</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"><font color="Maroon">c<sub>12</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <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>12</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self">Th23</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="E13:61"/><i><font color="Green">E63</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"><font color="Maroon">c<sub>13</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <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>13</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self">Th23</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="E15:61"/><i><font color="Green">E64</font></i>: 
( ( <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"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <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>14</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 <b>by </b><i><a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="afproj.html#E6:61"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E17:61"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</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="E19:61"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E20:61"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a>, <a class="ref" href="aff_4.html#T38">AFF_4:38</a></i>;<br/>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E69</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"><font color="Maroon">c<sub>13</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><a NAME="E2:61_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="afproj.html#E5:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E7:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E17:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E19:61"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E63"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1"/><i><font color="Green">E72</font></i>: 
( <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"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><a NAME="E2:61_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="afproj.html#E5:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E7:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E17:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E19:61"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E63"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E3:61_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#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E1:61_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E2:61_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="afproj.html#E1:61_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2"/><i><font color="Green">E72</font></i>: 
( <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>14</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/><a NAME="E2:61_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="afproj.html#E5:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E7:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E17:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E19:61"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E64"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>13</sub></font> 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="txt" href="afproj.html#E1:61_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E4:61_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="afproj.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E5:61_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E1:61_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E2:61_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_4.html#T42">AFF_4:42</a></i>;<br/><a NAME="E6:61_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E2:61_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="afproj.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E7:61_1_2"/>
<font color="Maroon">c<sub>18</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#E1:61_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E1:61_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:61_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#E15:61"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E3:61_1"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2"/><i><font color="Green">E69</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>13</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/><a NAME="E2:61_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="afproj.html#E5:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E7:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E17:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E19:61"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E64"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1"/><i><font color="Green">E72</font></i>: 
( <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>14</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/><a NAME="E2:61_2_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="afproj.html#E5:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E7:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E17:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E19:61"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E64"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E3:61_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E1:61_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E2:61_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="afproj.html#E1:61_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="aff_4.html#T61">AFF_4:61</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</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><b> such that </b><br/><a NAME="E5:61_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E6:61_2_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E57"><i><font color="Green">Lemma44</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E7:61_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:61_2_1"><i><font color="Green">E74</font></i></a></i>;<br/><b>thus </b><a NAME="E8:61_2_1"/>
<font color="Maroon">c<sub>20</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#E1:61_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E1:61_2_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="afproj.html#E3:61_2_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="afproj.html#E5:61_2_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afproj.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_2"/><i><font color="Green">E72</font></i>: 
( <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"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><a NAME="E2:61_2_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="afproj.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="afproj.html#E5:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="afproj.html#E7:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E9:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E11:61"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E17:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E19:61"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E63"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>14</sub></font> 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="txt" href="afproj.html#E1:61_2_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><a NAME="E4:61_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E1:61_2_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="afproj.html#E2:61_2_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="aff_4.html#T42">AFF_4:42</a></i>;<br/><a NAME="E5:61_2_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E20:61"><i><font color="Green">E67</font></i></a>, <a class="txt" href="afproj.html#E2:61_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="afproj.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E6:61_2_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:61_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E1:61_2_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T29" target="_self">Th29</a></i>;<br/><b>thus </b><a NAME="E7:61_2_2"/>
<font color="Maroon">c<sub>18</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#E1:61_2_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:61_2"/>
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#E15:61"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E3:61_2"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E23:61"/>
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#E13:61"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E21:61"><i><font color="Green">E68</font></i></a></i>;<br/>


</div>
