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<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">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">c<sub>2</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>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:24_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E5:24_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</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="E7:24_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E8:24_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E3:24_1"><i><font color="Green">E20</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E9:24_1"/>
<font color="Maroon">c<sub>2</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>6</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E4:24_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="afproj.html#E5:24_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="afproj.html#E7:24_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E10:24_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E8:24_1"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><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="E1:24_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</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>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:24_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 ;<br/><a NAME="E3:24_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:24_2"><i><font color="Green">E21</font></i></a></i>;<br/><b>hence </b><a NAME="E4:24_2"/>
<font color="Maroon">c<sub>2</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#E1:24_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
( <font color="Maroon">c<sub>2</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> iff 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> <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="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#E1:24"><i><font color="Green">E19</font></i></a></i>;<br/>


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