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<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 <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c2">F</font></span>)</span>; <span class="comment"><font color="firebrick">:: 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="firebrick">::  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="E1:40_2_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="firebrick">::  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="E2:40_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#K2" title="SIMPLEX0:func.2">subset-closed_closure_of</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:40_2_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E5:40_2_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T2" title="SIMPLEX0:th.2">SIMPLEX0:2</a></span>;<br/>
<a NAME="E6:40_2_1"/>
<font color="Maroon" title="c4">x</font> is   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_2_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="rlaffin1.html#D5" title="RLAFFIN1:def.5">RLAFFIN1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40_2_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_2_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="rlaffin1.html#T43" title="RLAFFIN1:th.43">RLAFFIN1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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