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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">K</font>, <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">X</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">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b2">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b3">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">M</font> holds <br/>( <font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b4">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  iff <font color="Olive" title="b4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">X</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c2">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">M</font> implies ( <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  iff <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c2">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c2">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">M</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">K</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  iff <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E2:59_1"/><b>then </b>
 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">K</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">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">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> )
 <b>by </b><i><a class="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E3:59_1"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:59"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="aff_4.html#T26" target="_self" title="AFF_4:th.26">Th26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:59"/>
( <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">A</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  iff <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="#E2:59"><i><font color="Green" title="E50">A2</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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