<?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>assume </b><a NAME="E1:75"/><i><font color="Green">E60</font></i>: 
 <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font> is <a href="incproj.html#V10">Desarguesian</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:75"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font>, <font color="Olive">b<sub>9</sub></font>, <font color="Olive">b<sub>10</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font> &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>10</sub></font> holds <br/><font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>9</sub></font>,<font color="Olive">b<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:75_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E2:75_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E3:75_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:75_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E5:75_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>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>and </b><br/><a NAME="E6:75_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1"/><i><font color="Green">E68</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>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><a NAME="E2:75_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E5:75_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E6:75_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="aff_4.html#T8">AFF_4:8</a>, <a class="ref" href="aff_4.html#T9">AFF_4:9</a></i>;<br/><a NAME="E3:75_1_1_1"/><b>then </b><i><font color="Green">E70</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> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E5:75_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E1:75_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E4:75_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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>8</sub></font>,<font color="Maroon">c<sub>9</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>7</sub></font>,<font color="Maroon">c<sub>9</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>10</sub></font>,<font color="Maroon">c<sub>11</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>11</sub></font>,<font color="Maroon">c<sub>12</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>10</sub></font>,<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="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E5:75_1_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E6:75_1_1_1"/><i><font color="Green">E73</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E6:75_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="afproj.html#E3:75_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="aff_1.html#T51">AFF_1:51</a></i>;<br/><a NAME="E7:75_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>12</sub></font> 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="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</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#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">c<sub>28</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>,1</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#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self">Th23</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>29</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>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>, <font color="Maroon">c<sub>30</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>7</sub></font>,<font color="Maroon">c<sub>9</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#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E12:75_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E13:75_1_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/><div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E3:75_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1_1"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:75_1_1_1"/>
( not <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>26</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>26</sub></font> )
 ;<br/>

<a NAME="E2:75_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E2:75_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="afproj.html#E13:75_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:75_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E2:75_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="afproj.html#E14:75_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being    <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> such that </b><br/><a NAME="E17:75_1_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="afproj.html#E13:75_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="afproj.html#E14:75_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="incproj.html#D13">INCPROJ:def 13</a></i>;<br/><a NAME="E18:75_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E19:75_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E12:75_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><a NAME="E20:75_1_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a>, <a class="ref" href="afproj.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E21:75_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="afproj.html#E7:75_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a>, <a class="ref" href="afproj.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E22:75_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>31</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E17:75_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="afproj.html#E20:75_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E3:75_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_4"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E3:75_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E25:75_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="incproj.html#R1">are_mutually_different</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E14:75_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="afproj.html#E23:75_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>32</sub></font> being    <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> such that </b><br/><a NAME="E27:75_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>31</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:75"><i><font color="Green">E60</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E1:75_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="afproj.html#E19:75_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="afproj.html#E22:75_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="incproj.html#D19">INCPROJ:def 19</a></i>;<br/><a NAME="E28:75_1_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>32</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E27:75_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/><div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_5"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E2:75_1_1_1_5"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:75_1_1_1_5"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:75_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E4:75_1_1_1_5"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:75_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E5:75_1_1_1_5"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="afproj.html#E2:75_1_1_1_5"><i><font color="Green">E86</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><b>hence </b><a NAME="E6:75_1_1_1_5"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="afproj.html#E6:75_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_1_6"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E2:75_1_1_1_6"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>32</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="Olive">b<sub>1</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>33</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>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>, <font color="Maroon">c<sub>34</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>7</sub></font>,<font color="Maroon">c<sub>9</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#K14" target="_self">ProjHorizon</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="afproj.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E2:75_1_1_1_6_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>32</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U3">Inc</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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>32</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U3">Inc</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#E28:75_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="incsp_1.html#D1">INCSP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>35</sub></font> being    <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#K14" target="_self">ProjHorizon</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:75_1_1_1_6_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>32</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>35</sub></font>,2</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1_1_1_6"><i><font color="Green">E86</font></i></a>, <a class="ref" href="afproj.html#T41" target="_self">Th41</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>36</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="E6:75_1_1_1_6_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T24" target="_self">Th24</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>36</sub></font>
;<br/>

<b>thus </b><a NAME="E7:75_1_1_1_6_1"/>
( <font color="Maroon">c<sub>32</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>36</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1_6_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="afproj.html#E6:75_1_1_1_6_1"><i><font color="Green">E88</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:75_1_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> is  not   <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#E28:75_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self">Th27</a></i>;<br/><b>then 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="E5:75_1_1_1_6"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E6:75_1_1_1_6"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E17:75_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1_6"><i><font color="Green">E87</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E7:75_1_1_1_6"/><b>then </b>
( <font color="Maroon">c<sub>33</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:75_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="afproj.html#E5:75_1_1_1_6"><i><font color="Green">E87</font></i></a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/><a NAME="E8:75_1_1_1_6"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E9:75_1_1_1_6"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:75_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E31:75_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E29:75_1_1_1"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:75_1_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:75_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="afproj.html#E2:75_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="afproj.html#E3:75_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="afproj.html#E5:75_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="afproj.html#E6:75_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="afproj.html#T50" target="_self">Th50</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:75_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:75_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="afproj.html#E4:75_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:75"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="aff_2.html#V4">Desarguesian</a>
 <b>by </b><i><a class="ref" href="aff_2.html#D4">AFF_2:def 4</a></i>;<br/>


</div>
