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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="simplex0.html#V5" title="SIMPLEX0:attr.5">with_non-empty_elements</a>  implies  not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1_1"/>
not <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V5" title="SIMPLEX0:attr.5">with_non-empty_elements</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> </span><br/>

<a NAME="E2:54_1_1"/><span class="kw">then </span>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#NV1" title="SIMPLEX0:NV.1">with_empty_element</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D8" target="_self" title="SIMPLEX0:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54_1_1"/>
not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D4" title="PENCIL_1:def.4">PENCIL_1:def 4</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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