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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E55">Th68</font></span>: <a NAME="T68"><span class="comment"><font color="firebrick">:: GTARSKI5:68</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="gtarski1.html#V9" title="GTARSKI1:attr.9">satisfying_Tarski-model</a>   <a href="gtarski2.html#NV5" title="GTARSKI2:NV.5">satisfying_Lower_Dimension_Axiom</a>   <a href="gtarski1.html#L1" title="GTARSKI1:struct.1">TarskiGeometryStruct</a> <br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">d</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Olive" title="b1">S</font> holds <br/> ( <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</font> <a href="gtarski5.html#R10" title="GTARSKI5:pred.10">are_coplanar</a>  iff  ex <font color="Olive" title="b6">x</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Olive" title="b1">S</font> st <br/>( (  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b6">x</font> &amp;  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</font>,<font color="Olive" title="b6">x</font> ) or (  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">x</font> &amp;  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">d</font>,<font color="Olive" title="b6">x</font> ) or (  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">d</font>,<font color="Olive" title="b6">x</font> &amp;  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">x</font> ) ) )</div></div>
