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







<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Q</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Q</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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_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="E1:27_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">Q</font>
 ;<br/><a NAME="E2:27_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">Q</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:27_1"/><i><font color="Green">E37</font></i>: 
 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="Maroon">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">Q</font> )
 ;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E7:27_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E8:27_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E10:27_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E11:27_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E13:27_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E14:27_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E15:27_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><a NAME="E16:27_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/><a NAME="E17:27_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E18:27_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><b>hence </b><a NAME="E19:27_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>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:27"/>
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">Q</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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