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<span class="kw">let </span><font color="Maroon" title="c1">k</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">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p2">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p3">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b2">Aff</font>  st ( for <font color="Olive" title="b4">v</font> being  <a href="simplex0.html#NM2" title="SIMPLEX0:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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="Olive" title="b2">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><br/> for <font color="Olive" title="b5">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Aff</font> &amp; <font color="Olive" title="b4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b5">B</font> holds <br/><font color="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">B</font> ) holds <br/> ex <font color="Olive" title="b4">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Aff</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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> st <font color="Olive" title="b3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Aff</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p2">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p3">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">Aff</font>  st ( for <font color="Olive" title="b3">v</font> being  <a href="simplex0.html#NM2" title="SIMPLEX0:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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="Olive" title="b1">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><br/> for <font color="Olive" title="b4">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Aff</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b4">B</font> holds <br/><font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">B</font> ) holds <br/> ex <font color="Olive" title="b3">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Aff</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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> st <font color="Olive" title="b2">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Aff</font></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">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p2">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p3">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">Aff</font>  st ( for <font color="Olive" title="b2">v</font> being  <a href="simplex0.html#NM2" title="SIMPLEX0:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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><br/> for <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Aff</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b3">B</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">B</font> ) holds <br/> ex <font color="Olive" title="b2">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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> st <font color="Olive" title="b1">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p2">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p3">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">Aff</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>,<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"> ( ( for <font color="Olive" title="b1">v</font> being  <a href="simplex0.html#NM2" title="SIMPLEX0:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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><br/> for <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b2">B</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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> ) implies  ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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> st <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">XX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font> </span>}</span> ;<br/>
<span class="kw">assume </span><a NAME="E1:74"/>
for <font color="Olive" title="b1">v</font> being  <a href="simplex0.html#NM2" title="SIMPLEX0:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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><br/> for <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b2">B</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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</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">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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> st <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font></span><br/>

<a NAME="E2:74"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">S</font> where <font color="Olive" title="b2">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T46" target="_self" title="SIMPLEX1:th.46">Th46</a></span>;<br/>
<a NAME="E3:74"/><span class="kw">then </span>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font> </span>}</span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:74"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E6:74"/>
 ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Aff</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:74"/>
 ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <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, <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<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> st <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> verum</span><br/>


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