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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">K</font> being non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex0.html#V6" title="SIMPLEX0:attr.6">total</a>   <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>   <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</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">Af</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b4">Sk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Olive" title="b2">K</font>  st <font color="Olive" title="b4">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b2">K</font> holds <br/>( ( <font color="Olive" title="b3">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Olive" title="b2">K</font> &amp; <span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b4">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Af</font> ) iff  ex <font color="Olive" title="b5">Tk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Olive" title="b2">K</font> st <br/>( <font color="Olive" title="b5">Tk</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">Sk</font> &amp; <font color="Olive" title="b5">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b5">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b5">Tk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b3">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b4">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b5">Tk</font></span>)</span> ) )</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">K</font> being non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex0.html#V6" title="SIMPLEX0:attr.6">total</a>   <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>   <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</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">Af</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b3">Sk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Olive" title="b1">K</font>  st <font color="Olive" title="b3">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">K</font> holds <br/>( ( <font color="Olive" title="b2">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Olive" title="b1">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Af</font> ) iff  ex <font color="Olive" title="b4">Tk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Olive" title="b1">K</font> st <br/>( <font color="Olive" title="b4">Tk</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Sk</font> &amp; <font color="Olive" title="b4">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b4">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">Tk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b2">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b4">Tk</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be  non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex0.html#V6" title="SIMPLEX0:attr.6">total</a>   <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>   <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</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">Af</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">Sk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font>  st <font color="Olive" title="b2">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">K</font> holds <br/>( ( <font color="Olive" title="b1">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Af</font> ) iff  ex <font color="Olive" title="b3">Tk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> st <br/>( <font color="Olive" title="b3">Tk</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Sk</font> &amp; <font color="Olive" title="b3">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b3">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">Tk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">Tk</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">Af</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">Sk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font>  st <font color="Olive" title="b1">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">K</font> holds <br/>( ( <font color="Maroon" title="c4">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Af</font> ) iff  ex <font color="Olive" title="b2">Tk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> st <br/>( <font color="Olive" title="b2">Tk</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Sk</font> &amp; <font color="Olive" title="b2">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b2">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Tk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">Tk</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">Sk</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</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="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">K</font> implies ( ( <font color="Maroon" title="c4">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Af</font> ) iff  ex <font color="Olive" title="b1">Tk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> st <br/>( <font color="Olive" title="b1">Tk</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font> &amp; <font color="Olive" title="b1">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Tk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Tk</font></span>)</span> ) ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">B</font> =  <a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c2">V</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">BK</font> =  <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a> 
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">K</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="c4">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Af</font> ) iff  ex <font color="Olive" title="b1">Tk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> st <br/>( <font color="Olive" title="b1">Tk</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font> &amp; <font color="Olive" title="b1">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Tk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Tk</font></span>)</span> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">nc</font> = <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span> as   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E4:68"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">nc</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">nc</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T22" title="XXREAL_3:th.22">XXREAL_3:22</a></span>;<br/>
<a NAME="E5:68"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D10" title="SIMPLEX0:def.10">SIMPLEX0:def 10</a></span>;<br/>
<a NAME="E6:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">K</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">K</font>
 
;<br/>
<a NAME="E7:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <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">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#D5" target="_self" title="SIMPLEX1:def.5">Def5</a></span>;<br/>
<a NAME="E8:68"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c8">nc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="simplex1.html#T31" target="_self" title="SIMPLEX1:th.31">Th31</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">Tk</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> st <br/>( <font color="Olive" title="b1">Tk</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font> &amp; <font color="Olive" title="b1">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">Tk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Tk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Tk</font></span>)</span> ) implies ( <font color="Maroon" title="c4">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Af</font> ) )</span>
<div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68_1"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
<font color="Maroon" title="c4">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E2:68_1"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Af</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font> &amp; <font color="Olive" title="b1">R</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">R</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">R</font></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c9">T</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E4:68_1"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
<font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a></span>;<br/><a NAME="E5:68_1"/>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">T</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T9" title="SIMPLEX0:th.9">SIMPLEX0:9</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/><a NAME="E6:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A11</font></span>: 
not  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">T</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/><a NAME="E7:68_1"/><span class="kw">then </span>
 <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">T</font></span>)</span> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">UT</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">T</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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="c2">V</font> ;<br/><a NAME="E9:68_1"/>
<font color="Maroon" title="c10">UT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="simplex1.html#K2" title="SIMPLEX1:func.2">@</a> <font color="Maroon" title="c9">T</font></span>)</span>
 ;<br/><a NAME="E10:68_1"/><span class="kw">then </span>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">Af</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c10">UT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="convex1.html#T30" title="CONVEX1:th.30">CONVEX1:30</a>, <a class="ref" href="rlaffin2.html#T17" title="RLAFFIN2:th.17">RLAFFIN2:17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">s1</font> = <font color="Maroon" title="c4">Af</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <span class="p2">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="simplex1.html#T40" target="_self" title="SIMPLEX1:th.40">Th40</a></span>;<br/><a NAME="E12:68_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">nc</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/><a NAME="E13:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c11">s1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="simplex0.html#T48" title="SIMPLEX0:th.48">SIMPLEX0:48</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">C</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">cT</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">UT</font> as   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><a NAME="E15:68_1"/>
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/><a NAME="E16:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A13</font></span>: 
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <span class="p2">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T31" target="_self" title="SIMPLEX1:th.31">Th31</a></span>;<br/><a NAME="E17:68_1"/>
(  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">cT</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p2">(<span class="default"><a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <span class="p3">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">UT</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T24" title="SIMPLEX0:th.24">SIMPLEX0:24</a>, <a class="ref" href="simplex0.html#T26" title="SIMPLEX0:th.26">SIMPLEX0:26</a></span>;<br/><a NAME="E18:68_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">UT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:68_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="ref" href="xxreal_3.html#T22" title="XXREAL_3:th.22">XXREAL_3:22</a></span>;<br/><a NAME="E19:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">nc</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">UT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/><a NAME="E20:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">UT</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>;<br/><a NAME="E21:68_1"/>
(  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c3">K</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a>  &amp; <font color="Maroon" title="c10">UT</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="c3">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><a NAME="E22:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A15</font></span>: 
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c10">UT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T1" title="SIMPLEX0:th.1">SIMPLEX0:1</a></span>;<br/><a NAME="E23:68_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="simplex1.html#K2" title="SIMPLEX1:func.2">@</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="simplex1.html#T34" target="_self" title="SIMPLEX1:th.34">Th34</a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">R</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E25:68_1"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
( <font color="Maroon" title="c14">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font> &amp; <font color="Maroon" title="c14">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Maroon" title="c14">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E26:68_1"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c14">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">UT</font>
 <span class="kw">and </span><br/><a NAME="E27:68_1"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E13:68_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E20:68_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="ref" href="simplex1.html#T35" target="_self" title="SIMPLEX1:th.35">Th35</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">R</font> = <font color="Maroon" title="c14">R</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>;<br/><a NAME="E29:68_1"/>
( <font color="Maroon" title="c15">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c15">R</font></span>)</span> &amp;  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c15">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c10">UT</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:68_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="simplex0.html#T1" title="SIMPLEX0:th.1">SIMPLEX0:1</a>, <a class="ref" href="zfmisc_1.html#T100" title="ZFMISC_1:th.100">ZFMISC_1:100</a></span>;<br/><a NAME="E30:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c10">UT</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E31:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:68_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">R</font> = <font color="Maroon" title="c15">R</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T14" title="SIMPLEX0:th.14">SIMPLEX0:14</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c17">R</font> = <font color="Maroon" title="c16">R</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="c17">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font> &amp; <font color="Maroon" title="c17">R</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Maroon" title="c17">R</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">R</font></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E33:68_1"/>
( <font color="Maroon" title="c17">R</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font> &amp; <font color="Maroon" title="c17">R</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Maroon" title="c17">R</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">R</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:68_1"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E27:68_1"><span class="lab"><font color="Green" title="E63">A18</font></span></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/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c9">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><a NAME="E11:68"/><span class="lab"><font color="Green" title="E53">A19</font></span>: 
<font color="Maroon" title="c9">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Sk</font>
 <span class="kw">and </span><br/><a NAME="E12:68"/><span class="lab"><font color="Green" title="E54">A20</font></span>: 
( <font color="Maroon" title="c9">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Maroon" title="c9">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Sk</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  )
 <span class="kw">and </span><br/><a NAME="E13:68"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">and </span><br/><a NAME="E14:68"/><span class="lab"><font color="Green" title="E56">A22</font></span>: 
<font color="Maroon" title="c4">Af</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">T</font></span>)</span>
 ; <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="c4">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Af</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">ST</font> = <font color="Maroon" title="c5">Sk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">T</font>;<br/>
<a NAME="E15:68"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">BST</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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Sk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">T</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D10" title="SIMPLEX0:def.10">SIMPLEX0:def 10</a></span>;<br/>
<a NAME="E17:68"/><span class="lab"><font color="Green" title="E57">A23</font></span>: 
<font color="Maroon" title="c5">Sk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">T</font> is  <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T20" title="TOPS_2:th.20">TOPS_2:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">BST</font> = <font color="Maroon" title="c11">BST</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a></span>;<br/>
<a NAME="E19:68"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Sk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E53">A19</font></span></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></span>;<br/>
<a NAME="E20:68"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">BST</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="txt" href="#E13:68"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E17:68"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="simplex1.html#T33" target="_self" title="SIMPLEX1:th.33">Th33</a></span>;<br/>
<a NAME="E21:68"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Sk</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">T</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">BST</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">nc</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:68"/>
( <font color="Maroon" title="c4">Af</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c1">n</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Sk</font></span>)</span>, <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <font color="Maroon" title="c3">K</font> &amp; <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> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">Sk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Af</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E14:68"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="simplex0.html#T48" title="SIMPLEX0:th.48">SIMPLEX0:48</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</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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