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<span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="simplex1.html#NM1" title="SIMPLEX1:NM.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  implies <font color="Maroon" title="c2">K</font> is  <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_1"/>
<font color="Maroon" title="c2">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">K</font> is  <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a> </span><br/>

<a NAME="E2:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A1</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">K</font> is  <a href="setfam_1.html#V2" title="SETFAM_1:attr.2">empty-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D7" title="SIMPLEX0:def.7">SIMPLEX0:def 7</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="simplex1.html#D7" target="_self" title="SIMPLEX1:def.7">SIMPLEX1:def 7</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="c3">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  implies  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:40_1_1"/>
not <font color="Maroon" title="c3">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> </span><br/>

<a NAME="E4:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E5:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_1_1"/>
 <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 ; <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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