<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font>, <font color="Olive" title="b3">K</font>, <font color="Olive" title="b4">A</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">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">N</font>,<font color="Olive" title="b3">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">N</font>,<font color="Olive" title="b4">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> holds <br/><font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">K</font>,<font color="Olive" title="b4">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</font>, <font color="Maroon" title="c4">K</font>, <font color="Maroon" title="c5">A</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> implies <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
( <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  )
 <span class="kw">and </span><br/><a NAME="E2:58"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 <span class="kw">and </span><br/><a NAME="E3:58"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 <span class="kw">and </span><br/><a NAME="E4:58"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> </span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:58"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E7:58"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<font color="Maroon" title="c3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E8:58"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<font color="Maroon" title="c4">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E9:58"/><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<font color="Maroon" title="c6">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></span>;<br/>
<a NAME="E10:58"/>
 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">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="c3">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="c5">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> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></span>;<br/>
<a NAME="E11:58"/><span class="kw">then </span>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E7:58"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E9:58"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="aff_4.html#T26" target="_self" title="AFF_4:th.26">Th26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:58"/>
<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E9:58"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
