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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">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b2">K</font>, <font color="Olive" title="b3">M</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">M</font> &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>  holds <br/>( <font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">M</font>,<font color="Olive" title="b3">M</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="b2">K</font>,<font color="Olive" title="b3">M</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="b3">M</font>,<font color="Olive" title="b2">K</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">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">M</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="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b1">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/>( <font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">M</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="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="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/><b>let </b><font color="Maroon" title="c3">K</font>, <font color="Maroon" title="c4">M</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &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>  implies ( <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</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="c3">K</font>,<font color="Maroon" title="c4">M</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="c4">M</font>,<font color="Maroon" title="c3">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  ) )</span><br/>







<b>assume </b><a NAME="E1:61"/>
( <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</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="c3">K</font>,<font color="Maroon" title="c4">M</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="c4">M</font>,<font color="Maroon" title="c3">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  )</span><br/>

<a NAME="E2:61"/><b>then </b>
 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="c3">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="c4">M</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> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 
<b>by </b><i><a class="ref" href="aff_4.html#T38" target="_self" title="AFF_4:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E3:61"/>
( <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</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="c3">K</font>,<font color="Maroon" title="c4">M</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="c4">M</font>,<font color="Maroon" title="c3">K</font> <a href="aff_4.html#R2" title="AFF_4:pred.2">is_coplanar</a>  )
 <b>by </b><i><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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