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<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">K</font>, <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">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">K</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">N</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  holds <br/>( <font color="Olive" title="b1">K</font>,<font color="Olive" title="b3">N</font>,<font color="Olive" title="b2">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b3">N</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">N</font>,<font color="Olive" title="b1">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b3">N</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Olive" title="b3">N</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">K</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">K</font>, <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">N</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  implies ( <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c3">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">N</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c2">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:57"/>
<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">N</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a> 
 ; <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">K</font>,<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c3">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">N</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c2">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  )</span><br/>

<a NAME="E2:57"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">X</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">X</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">X</font> &amp; <font color="Maroon" title="c4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">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="ref" href="aff_4.html#D5" target="_self" title="AFF_4:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:57"/>
( <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c3">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">N</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c2">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">M</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  &amp; <font color="Maroon" title="c4">N</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">K</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="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/>


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