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<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/>





<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:82"/><i><font color="Green" title="E68">A2</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>and </b><br/><a NAME="E4:82"/><i><font color="Green" title="E69">A3</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T34" target="_self" title="AFF_4:th.34">Th34</a></i>;<br/>
<a NAME="E5:82"/><i><font color="Green" title="E70">A4</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">r</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E4:82"><i><font color="Green" title="E69">A3</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>;<br/>
<b>set </b><font color="Maroon" title="c8">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font>;<br/>
<a NAME="E6:82"/><i><font color="Green" title="E71">A5</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<a NAME="E7:82"/><i><font color="Green" title="E72">A6</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="txt" href="aff_4.html#E5:82"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">M'</font> = <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">N'</font> = <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font></span>)</span>;<br/>
<a NAME="E8:82"/><i><font color="Green" title="E73">A7</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font></span>)</span> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font></span>)</span> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E7:82"><i><font color="Green" title="E72">A6</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a>, <a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E10:82"/><i><font color="Green" title="E74">A8</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Y</font> )
 <b>and </b><br/><a NAME="E11:82"/><i><font color="Green" title="E75">A9</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T38" target="_self" title="AFF_4:th.38">Th38</a></i>;<br/>
<a NAME="E12:82"/><i><font color="Green" title="E76">A10</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:82"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:82"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="aff_4.html#E5:82"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/>
<a NAME="E13:82"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font>
 ;<br/>

<a NAME="E2:82_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E6:82"><i><font color="Green" title="E71">A5</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E3:82_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E4:82"><i><font color="Green" title="E69">A3</font></i></a>, <a class="txt" href="aff_4.html#E6:82"><i><font color="Green" title="E71">A5</font></i></a>, <a class="txt" href="aff_4.html#E7:82"><i><font color="Green" title="E72">A6</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:82"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c11">Y</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:82"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="aff_4.html#E8:82"><i><font color="Green" title="E73">A7</font></i></a>, <a class="txt" href="aff_4.html#E10:82"><i><font color="Green" title="E74">A8</font></i></a>, <a class="txt" href="aff_4.html#E11:82"><i><font color="Green" title="E75">A9</font></i></a>, <a class="txt" href="aff_4.html#E12:82"><i><font color="Green" title="E76">A10</font></i></a>, <a class="ref" href="aff_4.html#T55" target="_self" title="AFF_4:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E15:82"/>
 ex <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E8:82"><i><font color="Green" title="E73">A7</font></i></a>, <a class="txt" href="aff_4.html#E10:82"><i><font color="Green" title="E74">A8</font></i></a>, <a class="txt" href="aff_4.html#E11:82"><i><font color="Green" title="E75">A9</font></i></a></i>;<br/>


</div>
