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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">K</font>, <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b3">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b4">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b3">M</font>,<font color="Olive" title="b4">N</font>,<font color="Olive" title="b2">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b3">M</font>,<font color="Olive" title="b4">N</font>,<font color="Olive" title="b1">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">N</font> holds <br/><font color="Olive" title="b3">M</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b1">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> </span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">K</font>, <font color="Maroon" title="c4">M</font>, <font color="Maroon" title="c5">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c3">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c2">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">N</font> implies <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:59"/>
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E2:59"/>
<font color="Maroon" title="c3">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c4">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c3">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 <b>and </b><br/><a NAME="E5:59"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c2">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 <b>and </b><br/><a NAME="E6:59"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> </span><br/>

<b>consider </b><font color="Maroon" title="c6">X</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="E8:59"/><i><font color="Green" title="E52">A5</font></i>: 
( <font color="Maroon" title="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> )
 <b>and </b><br/><a NAME="E9:59"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 <b>and </b><br/><a NAME="E10:59"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c6">X</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#E4:59"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></i>;<br/>
<a NAME="E11:59"/>
 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="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#E5:59"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></i>;<br/>
<a NAME="E12:59"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:59"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="aff_4.html#E6:59"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="aff_4.html#E8:59"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="aff_4.html#E10:59"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="aff_4.html#T26" target="_self" title="AFF_4:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E13:59"/>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E8:59"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="aff_4.html#E9:59"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="aff_4.html#E10:59"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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