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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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><a NAME="E1:24"/><i><font color="Green">E19</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>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</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#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma14</font></i></a></i>;<br/><a NAME="E4:24_1"/><i><font color="Green">E22</font></i>: 
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</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>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_4.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E5:24_1"/><i><font color="Green">E23</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="aff_4.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E6:24_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E3:24_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/><a NAME="E7:24_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_4.html#E5:24_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="aff_4.html#E1:24_1"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E9:24_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E5:24_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E7:24_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><b>hence </b><a NAME="E10:24_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:24_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_4.html#E4:24_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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