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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>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E26</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>  )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="E5:30"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E6:30"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:30"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:30"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_4.html#D2" target="_self">Def2</a></i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:30_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_4.html#E3:30"><i><font color="Green">E27</font></i></a>, <a class="txt" href="aff_4.html#E5:30"><i><font color="Green">E28</font></i></a>, <a class="txt" href="aff_4.html#E6:30"><i><font color="Green">E29</font></i></a>, <a class="txt" href="aff_4.html#E7:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_4.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:30_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</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#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_4.html#E5:30"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_4.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:30_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_4.html#E3:30"><i><font color="Green">E27</font></i></a>, <a class="txt" href="aff_4.html#E5:30"><i><font color="Green">E28</font></i></a>, <a class="txt" href="aff_4.html#E6:30"><i><font color="Green">E29</font></i></a>, <a class="txt" href="aff_4.html#E7:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_4.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E3:30_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>3</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_4.html#E5:30"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_4.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:30_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</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#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:30"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</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#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E8:30"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>


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