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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x</font> is   <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">AS</font></span>)</span>
 ;<br/><a NAME="E2:34_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> 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">AS</font></span>)</span>
 ;<br/><a NAME="E3:34_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">AS</font>
 ;<br/><a NAME="E4:34_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</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">AS</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> <a href="xboole_0.html#K2">\/</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">AS</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>
 ;<br/><a NAME="E5:34_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</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">AS</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> implies  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <font color="Maroon">x</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">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) ) )
 <b>by </b><i/>;<br/><a NAME="E6:34_1"/>
( <font color="Maroon">x</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">AS</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> implies  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <font color="Maroon">x</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">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:34_1"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <font color="Maroon">x</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">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 <b>by </b><i><a class="txt" href="afproj.html#E1:34"><i><font color="Green">E25</font></i></a>, <a class="txt" href="afproj.html#E4:34_1"><i><font color="Green">E26</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="34_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><a NAME="E1:34_2"/><i><font color="Green">E31</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <font color="Maroon">x</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">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 ;<br/><div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><a NAME="E2:34_2_1"/><b>then </b>
<font color="Maroon">x</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">AS</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/>;<br/><a NAME="E3:34_2_1"/><b>then </b>
<font color="Maroon">x</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">AS</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> <a href="xboole_0.html#K2">\/</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">AS</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="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:34_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">AS</font>
 ;<br/><b>hence </b><a NAME="E5:34_2_1"/>
<font color="Maroon">x</font> is   <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">AS</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2_2"/>
( <font color="Maroon">x</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">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/><a NAME="E2:34_2_2"/><b>then </b>
<font color="Maroon">x</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">AS</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/>;<br/><a NAME="E3:34_2_2"/><b>then </b>
<font color="Maroon">x</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">AS</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> <a href="xboole_0.html#K2">\/</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">AS</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="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:34_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K10" target="_self">ProjectiveLines</a> <font color="Maroon">AS</font>
 ;<br/><b>hence </b><a NAME="E5:34_2_2"/>
<font color="Maroon">x</font> is   <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">AS</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:34_2"/>
<font color="Maroon">x</font> is   <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">AS</font></span>)</span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:34"><i><font color="Green">E25</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
( <font color="Maroon">x</font> is   <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">AS</font></span>)</span> iff  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) or ( <font color="Maroon">x</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">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) ) )
 <b>by </b><i><a class="ref" href="afproj.html#T1" target="_self">Th1</a></i>;<br/>


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