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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">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> ;<br/><a NAME="E3:19_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T47">AFF_1:47</a></i>;<br/><a NAME="E4:19_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> :  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">a'</font>,<font color="Olive">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:19_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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