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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="E1:47"/>
<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="E3:47"/><i><font color="Green" title="E41">A1</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="E4:47"/><i><font color="Green" title="E42">A2</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E43">A3</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c4">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c5">q</font> )
 ;<br/><a NAME="E2:47_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><a NAME="E3:47_1"/><b>then </b><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:47"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="aff_4.html#E1:47_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/><b>take </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c4">p</font>;<br/><b>take </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c5">q</font>;<br/><b>thus </b><a NAME="E4:47_1"/>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:47"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E5:47_1"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:47"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="aff_4.html#E3:47_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:47"/>
 ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:47"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>


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