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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">Aff</font> being  <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="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">Bf</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">S</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="Olive" title="b1">V</font>  st <font color="Olive" title="b4">S</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Aff</font> &amp; <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="Olive" title="b4">S</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="Olive" title="b2">Aff</font> holds <br/>( ( <font color="Olive" title="b3">Bf</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">S</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="Olive" title="b2">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Bf</font> ) iff  ex <font color="Olive" title="b5">T</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="Olive" title="b1">V</font> st <br/>( <font color="Olive" title="b5">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">S</font> &amp; <font color="Olive" title="b5">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b5">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b5">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Aff</font> &amp; <font color="Olive" title="b3">Bf</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">S</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">T</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">Aff</font> being  <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/> for <font color="Olive" title="b2">Bf</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">S</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>  st <font color="Olive" title="b3">S</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Aff</font> &amp; <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="Olive" title="b3">S</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="Olive" title="b1">Aff</font> holds <br/>( ( <font color="Olive" title="b2">Bf</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">S</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="Olive" title="b1">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Bf</font> ) iff  ex <font color="Olive" title="b4">T</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> st <br/>( <font color="Olive" title="b4">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">S</font> &amp; <font color="Olive" title="b4">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b4">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b4">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Aff</font> &amp; <font color="Olive" title="b2">Bf</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">S</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">T</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">Aff</font> be   <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>; <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">Bf</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">S</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>  st <font color="Olive" title="b2">S</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <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="Olive" title="b2">S</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="c3">Aff</font> holds <br/>( ( <font color="Olive" title="b1">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Bf</font> ) iff  ex <font color="Olive" title="b3">T</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> st <br/>( <font color="Olive" title="b3">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b3">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Olive" title="b1">Bf</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">S</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">T</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">Bf</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">S</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>  st <font color="Olive" title="b1">S</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <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="Olive" title="b1">S</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="c3">Aff</font> holds <br/>( ( <font color="Maroon" title="c4">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bf</font> ) iff  ex <font color="Olive" title="b2">T</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> st <br/>( <font color="Olive" title="b2">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b2">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Maroon" title="c4">Bf</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">S</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">T</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <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>; <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">S</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <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">S</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="c3">Aff</font> implies ( ( <font color="Maroon" title="c4">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bf</font> ) iff  ex <font color="Olive" title="b1">T</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> st <br/>( <font color="Olive" title="b1">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Maroon" title="c4">Bf</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">S</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">T</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">U</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">S</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c5">S</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="E2:61"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font>
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E41">A3</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">S</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="c3">Aff</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">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bf</font> ) iff  ex <font color="Olive" title="b1">T</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> st <br/>( <font color="Olive" title="b1">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Maroon" title="c4">Bf</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">S</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">T</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">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="c3">Aff</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="c9">c</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Aff</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/>
<span class="kw">set </span><font color="Maroon" title="c10">BTC</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#K5" title="RELSET_1:func.5">|</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">BC</font> =  <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<a NAME="E5:61"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c3">Aff</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="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c3">Aff</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T4" title="SIMPLEX0:th.4">SIMPLEX0:4</a></span>;<br/>
<span class="lab"><font color="Green" title="E43">A5</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> = 
<font color="Maroon" title="c9">c</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T26" title="SIMPLEX0:th.26">SIMPLEX0:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Aff</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
<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/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">c</font> = <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c1">n</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="E8:61"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<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="c3">Aff</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
;<br/>
<a NAME="E9:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A7</font></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="c3">Aff</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#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>,<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="c3">Aff</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="ref" href="simplex1.html#D5" target="_self" title="SIMPLEX1:def.5">Def5</a></span>;<br/>
<a NAME="E10:61"/>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">S</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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Aff</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E11:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A8</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">S</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> <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="c3">Aff</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="#E6:61"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E8:61"><span class="lab"><font color="Green" title="E44">A6</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">T</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> st <br/>( <font color="Olive" title="b1">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Maroon" title="c4">Bf</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">S</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">T</font></span>)</span> ) implies ( <font color="Maroon" title="c4">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bf</font> ) )</span>
<div class="add"><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c5">S</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 <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> or <font color="Maroon" title="c13">x</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 <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_2_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</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="c13">x</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 <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><br/>

<a NAME="E2:61_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <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="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E3:61_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E40">A2</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">hence </span><a NAME="E4:61_2_1"/>
<font color="Maroon" title="c13">x</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 <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="c3">Aff</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="#E5:61"><span class="lab"><font color="Green" title="E42">A4</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><a NAME="E2:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A10</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">S</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#K5" title="RELSET_1:func.5">|</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></span>;<br/><a NAME="E3:61_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:61_2"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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#K5" title="RELSET_1:func.5">|</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp; <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>;<br/><a NAME="E5:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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#K5" title="RELSET_1:func.5">|</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c3">Aff</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="#E1:61_2"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:61_2"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c4">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:61_2"/><span class="lab"><font color="Green" title="E51">A13</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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bf</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">T</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> st <br/>( <font color="Olive" title="b1">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Maroon" title="c4">Bf</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">S</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">T</font></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">a</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 <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:61_2"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
<font color="Maroon" title="c4">Bf</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="c13">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E6:61_2"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a></span>;<br/><a NAME="E10:61_2"/>
<font color="Maroon" title="c13">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">AA</font> = <font color="Maroon" title="c13">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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></span>)</span> as   <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 <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="c3">Aff</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="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E12:61_2"/><span class="lab"><font color="Green" title="E53">A15</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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c14">AA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_2"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E9:61_2"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="relat_1.html#T145" title="RELAT_1:th.145">RELAT_1:145</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">T</font> = <font color="Maroon" title="c14">AA</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">S</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">V</font> ;<br/><a NAME="E14:61_2"/><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<font color="Maroon" title="c14">AA</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 <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="c3">Aff</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="ref" href="simplex0.html#T14" title="SIMPLEX0:th.14">SIMPLEX0:14</a></span>;<br/><a NAME="E15:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A17</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="c14">AA</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#K5" title="RELSET_1:func.5">|</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">AA</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></span>;<br/><span class="lab"><font color="Green" title="E56">A18</font></span>: <font color="Maroon" title="c5">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">T</font> = 
<font color="Maroon" title="c14">AA</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c14">AA</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E5:61_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E12:61_2"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E15:61_2"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="ref" href="funct_1.html#T157" title="FUNCT_1:th.157">FUNCT_1:157</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/><a NAME="E17:61_2"/>
<font color="Maroon" title="c15">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">AA</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><a NAME="E18:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A19</font></span>: 
<font color="Maroon" title="c15">T</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="c3">Aff</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E14:61_2"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E19:61_2"/><span class="lab"><font color="Green" title="E58">A20</font></span>: 
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">AA</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E20:61_2"/><span class="kw">then </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="c13">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<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></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">AA</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T145" title="RELAT_1:th.145">RELAT_1:145</a>, <a class="ref" href="setfam_1.html#D9" title="SETFAM_1:def.9">SETFAM_1:def 9</a></span>;<br/><a NAME="E21:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A21</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Bf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">AA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_2"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="simplex1.html#T33" target="_self" title="SIMPLEX1:th.33">Th33</a></span>;<br/><span class="lab"><font color="Green" title="E60">A22</font></span>: <font color="Maroon" title="c4">Bf</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> <font color="Maroon" title="c14">AA</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_2"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="relat_1.html#T145" title="RELAT_1:th.145">RELAT_1:145</a></span>
<br/>.= 
<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">S</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="c15">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">T</font> = <font color="Maroon" title="c15">T</font> as   <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> ;<br/><span class="kw">take </span><font color="Maroon" title="c17">T</font> = <font color="Maroon" title="c16">T</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">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Maroon" title="c17">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Maroon" title="c17">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Maroon" title="c4">Bf</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">S</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">T</font></span>)</span> )</span><br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Bf</font> = 
<font color="Maroon" title="c12">c</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="#E11:61"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E6:61_2"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>
<br/>.= 
<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">S</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="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>
;<br/><a NAME="E25:61_2"/><span class="kw">then </span>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c3">Aff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font> &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">S</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> <span class="p2">(<span class="default"><font color="Maroon" title="c14">AA</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span> <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">S</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="#E16:61_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E21:61_2"><span class="lab"><font color="Green" title="E59">A21</font></span></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a>, <a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></span>;<br/><span class="kw">hence </span><a NAME="E26:61_2"/>
( <font color="Maroon" title="c17">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Maroon" title="c17">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Maroon" title="c17">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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">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 &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Maroon" title="c4">Bf</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">S</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">T</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E18:61_2"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="txt" href="#E19:61_2"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="txt" href="#E22:61_2"><span class="lab"><font color="Green" title="E60">A22</font></span></a>, <a class="ref" href="setfam_1.html#D9" title="SETFAM_1:def.9">SETFAM_1:def 9</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</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="c13">T</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><a NAME="E14:61"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c13">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">and </span><br/><a NAME="E15:61"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
( <font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  &amp; <font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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="E16:61"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">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="E17:61"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c13">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font>
 <span class="kw">and </span><br/><a NAME="E18:61"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<font color="Maroon" title="c4">Bf</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">S</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="c13">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">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bf</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">TS</font> = <font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">t</font> = <font color="Maroon" title="c13">T</font> as   <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> ;<br/>
<span class="lab"><font color="Green" title="E52">A28</font></span>:  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c14">TS</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">t</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="c5">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></span>
<br/>.= 
<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">S</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="#E16:61"><span class="lab"><font color="Green" title="E49">A25</font></span></a></span>
;<br/>
<a NAME="E22:61"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="c13">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<a NAME="E23:61"/><span class="kw">then </span>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E17:61"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E24:61"/><span class="kw">then </span>
( <font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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> <span class="p2">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</font></span>)</span></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="c3">Aff</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T79" title="ZFMISC_1:th.79">ZFMISC_1:79</a>, <a class="ref" href="zfmisc_1.html#T100" title="ZFMISC_1:th.100">ZFMISC_1:100</a></span>;<br/>
<a NAME="E25:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<font color="Maroon" title="c13">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">S</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 <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="c3">Aff</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="#E5:61"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E26:61"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
<font color="Maroon" title="c14">TS</font> is  <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</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="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">TS</font> or  not <font color="Maroon" title="c16">a</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:61_4"/>
(  not <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">TS</font> or  not <font color="Maroon" title="c16">a</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:61"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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>
<a NAME="E27:61"/>
 <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> <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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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="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="c3">Aff</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="#E9:61"><span class="lab"><font color="Green" title="E45">A7</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="c16">BTS</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> <font color="Maroon" title="c14">TS</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="c3">Aff</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="#E9:61"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E15:61"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E26:61"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a></span>;<br/>
<a NAME="E29:61"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c14">TS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</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">TS</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E26:61"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="simplex1.html#T33" target="_self" title="SIMPLEX1:th.33">Th33</a></span>;<br/>
<a NAME="E30:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A31</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">BTS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</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="#E21:61"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>;<br/>
<a NAME="E31:61"/>
<font color="Maroon" title="c16">BTS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Bf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:61"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:61"/>
( <font color="Maroon" title="c4">Bf</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">S</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="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &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">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bf</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E18:61"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E30:61"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</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/>


</div>
