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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><br/> for <font color="Olive" title="b2">Kv</font> being non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">Sv</font> being non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b2">Kv</font>  st <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b2">Kv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b2">Kv</font> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b3">Sv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b3">Sv</font> holds <br/> <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">Sv</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">Kv</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Kv</font> being non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">Sv</font> being non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b1">Kv</font>  st <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b1">Kv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">Kv</font> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b2">Sv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b2">Sv</font> holds <br/> <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">Sv</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">Kv</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Kv</font> be  non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Sv</font> being non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c3">Kv</font>  st <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">Kv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Kv</font> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b1">Sv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">Sv</font> holds <br/> <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">Sv</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Kv</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">S</font> be  non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c3">Kv</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">Kv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Kv</font> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c4">S</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">S</font> implies  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">S</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Kv</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:33"/>
( <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">Kv</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Kv</font> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c4">S</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">S</font> )
 ; <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#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">S</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Kv</font></span><br/>

<a NAME="E2:33"/><span class="kw">then </span>
(  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c2">V</font></span>)</span>,<font color="Maroon" title="c4">S</font> &amp;  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Kv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c2">V</font></span>)</span>,<font color="Maroon" title="c3">Kv</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#D6" target="_self" title="SIMPLEX1:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:33"/>
 <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">S</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Kv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T65" title="SIMPLEX0:th.65">SIMPLEX0:65</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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