<?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> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>2</sub></font> is   <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/><a NAME="E2:32_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> 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>
 ;<br/><a NAME="E3:32_1"/><b>then </b>
<font color="Maroon">c<sub>2</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>
 ;<br/><a NAME="E4:32_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</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> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E5:32_1"/>
( <font color="Maroon">c<sub>2</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> implies 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>2</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="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) )
 <b>by </b><i><a class="ref" href="afproj.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E6:32_1"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</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="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) )
 <b>by </b><i><a class="txt" href="afproj.html#E4:32_1"><i><font color="Green">E24</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="32_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</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="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) )
 ;<br/><a NAME="E2:32_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>3</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="E1:32_2_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/><a NAME="E2:32_2_1"/>
<font color="Maroon">c<sub>2</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#E1:32_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="afproj.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E3:32_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</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> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:32_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</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>
 ;<br/><b>hence </b><a NAME="E5:32_2_1"/>
<font color="Maroon">c<sub>2</sub></font> is   <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/></div><b>end;</b></div><b>hence </b><a NAME="E4:32_2"/>
<font color="Maroon">c<sub>2</sub></font> is   <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#E1:32_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="afproj.html#E2:32_2"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32"/>
( <font color="Maroon">c<sub>2</sub></font> is   <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> iff ( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</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="Maroon">c<sub>2</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="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) ) )
 <b>by </b><i><a class="txt" href="afproj.html#E1:32"><i><font color="Green">E23</font></i></a></i>;<br/>


</div>
