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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="simplex0.html#NM1" title="SIMPLEX0:NM.1">SimplicialComplexStr</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp; <font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  implies  not <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V5" title="SIMPLEX0:attr.5">with_non-empty_elements</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_4_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
( not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp; <font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V5" title="SIMPLEX0:attr.5">with_non-empty_elements</a> </span><br/>

<a NAME="E2:54_4_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54_4_1"/>
not <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V5" title="SIMPLEX0:attr.5">with_non-empty_elements</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_4_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="simplex0.html#D8" target="_self" title="SIMPLEX0:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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