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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( 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="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">b<sub>1</sub></font> ) ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <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_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</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>2</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#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) );<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="relset_1.html#M2">Relation</a> of  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font>, <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:28_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E3:28_1_1"/>
(  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> or 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="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or <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="Olive">b<sub>1</sub></font> ) ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <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_4.html#NR1" target="_self">is_plane</a>  &amp; <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="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</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>2</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#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="afproj.html#E2:28_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>

<b>assume </b><a NAME="E4:28_1_1"/><i><font color="Green">E21</font></i>: 
( 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="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or <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="Olive">b<sub>1</sub></font> ) ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <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_4.html#NR1" target="_self">is_plane</a>  &amp; <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="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</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>2</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#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<a NAME="E5:28_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:28_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</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>7</sub></font>,1</span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E2:28_1_1_1_1"/><i><font color="Green">E24</font></i>: 
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) or <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>7</sub></font> )
 ;<br/><a NAME="E3:28_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:28_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="afproj.html#T18" target="_self">Th18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_1_1_1"/>
<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>7</sub></font>
 ;<br/><a NAME="E2:28_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:28_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="afproj.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:28_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E2:28_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="afproj.html#E3:28_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:28_1_1_1_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <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>7</sub></font> &amp; <font color="Maroon">c<sub>6</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>8</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:28_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:28_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="afproj.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:28_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:28_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="afproj.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K9" target="_self">ProjectivePoints</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:28_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="afproj.html#E1:28_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:28_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="afproj.html#E4:28_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
