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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">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="b5">S</font> where <font color="Olive" title="b5">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="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> : <font color="Olive" title="b3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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="b4">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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">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="b4">S</font> where <font color="Olive" title="b4">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="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> : <font color="Olive" title="b2">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="b1">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="b3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</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">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="b3">S</font> where <font color="Olive" title="b3">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="Olive" title="b1">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="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="b2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">O</font> = 1 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">reconsider </span><font color="Maroon" title="c5">Z</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">A</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>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <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">A</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">A</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">A</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">A</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">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="b4">S</font> where <font color="Olive" title="b4">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="Olive" title="b1">A</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">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Olive" title="b2">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="b1">A</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="b3">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1;<br/>
<a NAME="E3:73"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">m</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">A</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"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 implies 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="c7">A</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="c7">A</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="c7">A</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="c7">A</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">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="b3">S</font> where <font color="Olive" title="b3">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Olive" title="b1">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="Maroon" title="c7">A</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="b2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E2:73_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <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="c7">A</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="c7">A</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="c7">A</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="c7">A</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">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="b3">S</font> where <font color="Olive" title="b3">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Olive" title="b1">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="Maroon" title="c7">A</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="b2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/>

<a NAME="E3:73_1"/>
not <font color="Maroon" title="c7">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:73_1"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a></span>;<br/>
<a NAME="E7:73_1"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Aa</font> = <font color="Maroon" title="c7">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T43" title="RLAFFIN1:th.43">RLAFFIN1:43</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">CAa</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">BCAa</font> =  <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<a NAME="E9:73_1"/>
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="c10">Aa</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="c10">Aa</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="kw">then </span><span class="lab"><font color="Green" title="E56">A6</font></span>:  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</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="c10">Aa</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T16" title="SIMPLEX0:th.16">SIMPLEX0:16</a></span>
<br/>.= 
<font color="Maroon" title="c10">Aa</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></span>
;<br/>
<a NAME="E11:73_1"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</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> <font color="Maroon" title="c2">V</font> )
 
;<br/>
<a NAME="E12:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A8</font></span>: 
 <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T24" target="_self" title="SIMPLEX1:th.24">Th24</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">CA</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">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="c7">A</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="c7">A</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="c7">A</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="c7">A</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="c14">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">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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><br/>

<span class="kw">assume </span><a NAME="E13:73_1"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
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="c7">A</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="c7">A</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="c14">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">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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><br/>

<span class="kw">set </span><font color="Maroon" title="c15">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</font> </span>}</span> ;<br/>
<a NAME="E14:73_1"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</font> </span>}</span>  <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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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="c16">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</font> </span>}</span>  or <font color="Maroon" title="c16">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_2"/>
<font color="Maroon" title="c16">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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="c16">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:73_1_2"/><span class="kw">then </span>
 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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:73_1_2"/>
<font color="Maroon" title="c16">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">XX</font> = <font color="Maroon" title="c16">XX</font> as   <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="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="c7">A</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="#E14:73_1"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="simplex0.html#T14" title="SIMPLEX0:th.14">SIMPLEX0:14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">BCA</font> =  <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<a NAME="E17:73_1"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
(  <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="c7">A</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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font> &amp; <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="c7">A</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> <font color="Maroon" title="c2">V</font> )
 
;<br/>
<a NAME="E18:73_1"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</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="E19:73_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c19">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_3"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c19">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">A</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="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c19">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:73_1_3"/>
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c19">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E1:73_1_3"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E20:73_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/>
<a NAME="E21:73_1"/><span class="kw">then </span>
 <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T30" title="SIMPLEX0:th.30">SIMPLEX0:30</a></span>;<br/>
<a NAME="E22:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A14</font></span>: 
 <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">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="c7">A</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="#E17:73_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E11:73_1"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="simplex1.html#T23" target="_self" title="SIMPLEX1:th.23">Th23</a></span>;<br/>
<a NAME="E23:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A15</font></span>: 
 <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T31" title="SIMPLEX0:th.31">SIMPLEX0:31</a></span>;<br/>
<a NAME="E24:73_1"/><span class="lab"><font color="Green" title="E65">A16</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="c7">A</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="c7">A</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="kw">then </span><span class="lab"><font color="Green" title="E66">A17</font></span>:  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</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="c7">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T16" title="SIMPLEX0:th.16">SIMPLEX0:16</a></span>
<br/>.= 
<font color="Maroon" title="c7">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></span>
;<br/>
<a NAME="E26:73_1"/><span class="lab"><font color="Green" title="E67">A18</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c7">A</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="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_5"/>
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_1"/><span class="lab"><font color="Green" title="E68">A19</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <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">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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><br/><div class="add"><a NAME="E2:73_1_5_1"/><span class="lab"><font color="Green" title="E69">A20</font></span>: 
<font color="Maroon" title="c4">O</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T7" title="XXREAL_3:th.7">XXREAL_3:7</a></span>;<br/><a NAME="E3:73_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A21</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="c7">A</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="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:73_1_5_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="ref" href="simplex0.html#T26" title="SIMPLEX0:th.26">SIMPLEX0:26</a></span>;<br/><a NAME="E4:73_1_5_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/><a NAME="E5:73_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A22</font></span>: 
 <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="c7">A</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#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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E3:73_1_5_1"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="ref" href="simplex1.html#T16" target="_self" title="SIMPLEX1:th.16">Th16</a>, <a class="ref" href="simplex1.html#T22" target="_self" title="SIMPLEX1:th.22">Th22</a></span>;<br/><a NAME="E6:73_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</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="c7">A</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="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">take </span>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/><a NAME="E7:73_1_5_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D1" title="ZFMISC_1:def.1">ZFMISC_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">A1</font> = <font color="Maroon" title="c7">A</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</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="c7">A</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="#E24:73_1"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><a NAME="E9:73_1_5_1"/><span class="lab"><font color="Green" title="E73">A24</font></span>: 
<font color="Maroon" title="c19">A1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> , <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:73_1_5_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E2:73_1_5_1"><span class="lab"><font color="Green" title="E69">A20</font></span></a>, <a class="ref" href="simplex0.html#T48" title="SIMPLEX0:th.48">SIMPLEX0:48</a></span>;<br/><a NAME="E10:73_1_5_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:73_1_5_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></span>;<br/><a NAME="E11:73_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A25</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E12:73_1_5_1"/><span class="kw">then </span>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T1" title="RLAFFIN1:th.1">RLAFFIN1:1</a></span>;<br/><a NAME="E13:73_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E13:73_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E25:73_1"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="txt" href="#E5:73_1_5_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a></span>;<br/><a NAME="E14:73_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A26</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1_5_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E15:73_1_5_1"/><span class="lab"><font color="Green" title="E76">A27</font></span>: 
<font color="Maroon" title="c17">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">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="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</font> or <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</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="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> , <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><span class="kw"> such that </span><br/><a NAME="E3:73_1_5_1_1"/><span class="lab"><font color="Green" title="E77">A28</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">S</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_1_1"/>
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c21">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:73_1_5_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E5:73_1_5_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a></span>;<br/>
<a NAME="E5:73_1_5_1_1"/><span class="lab"><font color="Green" title="E78">A29</font></span>: 
<font color="Maroon" title="c21">S</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="c7">A</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="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c21">S</font> = 
<font color="Maroon" title="c5">Z</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="#E3:73_1_5_1"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>
;<br/>
<a NAME="E7:73_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E24:73_1"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="txt" href="#E1:73_1_5_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E5:73_1_5_1_1"><span class="lab"><font color="Green" title="E78">A29</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_5_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_1_1"><span class="lab"><font color="Green" title="E77">A28</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">A</font> = 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c14">F</font>,<font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1_5_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="ref" href="relat_1.html#D16" title="RELAT_1:def.16">RELAT_1:def 16</a></span>
<br/>.= 
<font color="Maroon" title="c7">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E25:73_1"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="txt" href="#E6:73_1_5_1"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E11:73_1_5_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E14:73_1_5_1"><span class="lab"><font color="Green" title="E75">A26</font></span></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></span>
;<br/><a NAME="E17:73_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:73_1_5_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E9:73_1_5_1"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="txt" href="#E5:73_1_5_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a></span>;<br/><a NAME="E18:73_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:73_1_5_1"><span class="lab"><font color="Green" title="E76">A27</font></span></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E19:73_1_5_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </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="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2"/><span class="lab"><font color="Green" title="E68">A30</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <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">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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $2;<br/><span class="kw">set </span><font color="Maroon" title="c19">XXA</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">m1</font> = <font color="Maroon" title="c6">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 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">reconsider </span><font color="Maroon" title="c21">M</font> = <font color="Maroon" title="c6">m</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">reconsider </span><font color="Maroon" title="c22">cA</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</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="c23">YA</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> ;<br/><a NAME="E5:73_1_5_2"/><span class="lab"><font color="Green" title="E69">A31</font></span>: 
<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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span>  <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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">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="c24">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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span>  or <font color="Maroon" title="c24">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_1"/>
<font color="Maroon" title="c24">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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</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="c24">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:73_1_5_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font> &amp; <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:73_1_5_2_1"/>
<font color="Maroon" title="c24">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="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><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">YA</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">YA</font> = <font color="Maroon" title="c24">YA</font> as   <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="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="c7">A</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="#E5:73_1_5_2"><span class="lab"><font color="Green" title="E69">A31</font></span></a>, <a class="ref" href="simplex0.html#T14" title="SIMPLEX0:th.14">SIMPLEX0:14</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1;<br/><span class="kw">set </span><font color="Maroon" title="c26">Xm1</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c27">Xm</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c28">R1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><span class="kw"> such that </span><br/><a NAME="E9:73_1_5_2"/><span class="lab"><font color="Green" title="E70">A32</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">S</font> where <font color="Olive" title="b3">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">S</font> where <font color="Olive" title="b3">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">S</font> </span>}</span>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="relat_1.html#S1" title="RELAT_1:sch.1">RELAT_1:sch 1</a>();<br/></span><span class="kw">set </span><font color="Maroon" title="c29">DY</font> = <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font>;<br/><a NAME="E10:73_1_5_2"/><span class="lab"><font color="Green" title="E71">A33</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">XX</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">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="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font> or <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_2"/><span class="lab"><font color="Green" title="E72">A34</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</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="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_2"/><span class="lab"><font color="Green" title="E73">A35</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c31">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E4:73_1_5_2_2"/>
<font color="Maroon" title="c30">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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_2"><span class="lab"><font color="Green" title="E73">A35</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</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="E6:73_1_5_2_2"/><span class="lab"><font color="Green" title="E74">A36</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">S</font>
 <span class="kw">and </span><br/><a NAME="E7:73_1_5_2_2"/>
verum
 ;<br/>
<a NAME="E8:73_1_5_2_2"/>
not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">YA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2_2"><span class="lab"><font color="Green" title="E72">A34</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:73_1_5_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A37</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c32">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_2"><span class="lab"><font color="Green" title="E74">A36</font></span></a></span>;<br/>
<a NAME="E10:73_1_5_2_2"/>
<font color="Maroon" title="c31">y</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_2"><span class="lab"><font color="Green" title="E73">A35</font></span></a></span>;<br/>
<a NAME="E11:73_1_5_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A38</font></span>: 
 ex <font color="Olive" title="b1">W</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> st <br/>( <font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> )
 
;<br/>
<a NAME="E12:73_1_5_2_2"/>
<font color="Maroon" title="c31">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_2"><span class="lab"><font color="Green" title="E73">A35</font></span></a></span>;<br/>
<a NAME="E13:73_1_5_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c32">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_2"><span class="lab"><font color="Green" title="E74">A36</font></span></a>, <a class="txt" href="#E11:73_1_5_2_2"><span class="lab"><font color="Green" title="E76">A38</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<a NAME="E14:73_1_5_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Aa</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c32">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2_2"><span class="lab"><font color="Green" title="E75">A37</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E15:73_1_5_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c32">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></span>;<br/>
<a NAME="E16:73_1_5_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A39</font></span>: 
<font color="Maroon" title="c6">m</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c32">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E17:73_1_5_2_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c32">S</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <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="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E18:73_1_5_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c32">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E16:73_1_5_2_2"><span class="lab"><font color="Green" title="E77">A39</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:73_1_5_2_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E6:73_1_5_2_2"><span class="lab"><font color="Green" title="E74">A36</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c30">RDY</font> = <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span>;<br/><a NAME="E11:73_1_5_2"/><span class="lab"><font color="Green" title="E72">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:73_1_5_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">xy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_3"/><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<font color="Maroon" title="c31">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></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/>
<span class="kw">consider </span><font color="Maroon" title="c32">x</font>, <font color="Maroon" title="c33">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:73_1_5_2_3"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<font color="Maroon" title="c31">xy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">x</font>,<font color="Maroon" title="c33">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_3"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E6:73_1_5_2_3"/><span class="lab"><font color="Green" title="E75">A43</font></span>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_3"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="txt" href="#E5:73_1_5_2_3"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E7:73_1_5_2_3"/><span class="kw">then </span>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_5_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_3"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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><span class="lab"><font color="Green" title="E73">A44</font></span>: 2 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> 2</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex1.html#E30"><span class="lab"><font color="Green" title="E25">Lm1</font></span></a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T52" title="CARD_2:th.52">CARD_2:52</a></span>
;<br/><a NAME="E13:73_1_5_2"/><span class="lab"><font color="Green" title="E74">A45</font></span>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E14:73_1_5_2"/>
<font color="Maroon" title="c22">cA</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</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="E15:73_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A46</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <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="kw">consider </span><font color="Maroon" title="c31">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><span class="kw"> such that </span><br/><a NAME="E17:73_1_5_2"/><span class="lab"><font color="Green" title="E76">A47</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">S</font> where <font color="Olive" title="b3">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">S</font> </span>}</span>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">S</font> where <font color="Olive" title="b3">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="relat_1.html#S1" title="RELAT_1:sch.1">RELAT_1:sch 1</a>();<br/></span><a NAME="E18:73_1_5_2"/><span class="lab"><font color="Green" title="E77">A48</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c31">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c28">R1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E1:73_1_5_2_5"/><span class="lab"><font color="Green" title="E78">A49</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c32">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c32">z</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_5_1"/><span class="lab"><font color="Green" title="E79">A50</font></span>: 
<font color="Maroon" title="c32">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c32">z</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font></span><br/>

<a NAME="E2:73_1_5_2_5_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c32">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E3:73_1_5_2_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A51</font></span>: 
<font color="Maroon" title="c32">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>;<br/>
<a NAME="E4:73_1_5_2_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A52</font></span>: 
<font color="Maroon" title="c32">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E1:73_1_5_2_5_1"><span class="lab"><font color="Green" title="E79">A50</font></span></a></span>;<br/>
<a NAME="E5:73_1_5_2_5_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c32">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c32">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ] &amp; <font color="Maroon" title="c32">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E1:73_1_5_2_5_1"><span class="lab"><font color="Green" title="E79">A50</font></span></a>, <a class="txt" href="#E3:73_1_5_2_5_1"><span class="lab"><font color="Green" title="E80">A51</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:73_1_5_2_5_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c32">z</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E4:73_1_5_2_5_1"><span class="lab"><font color="Green" title="E81">A52</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c32">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c31">R</font>,<font color="Maroon" title="c28">R1</font><span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_5"/><span class="lab"><font color="Green" title="E79">A53</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> holds <br/><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E1:73_1_5_2_5"><span class="lab"><font color="Green" title="E78">A49</font></span></a>);<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_5_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_5_2_5_2"/>
( <font color="Maroon" title="c28">R1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c28">R1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_5_2_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_5_2_1"/><span class="lab"><font color="Green" title="E80">A54</font></span>: 
<font color="Maroon" title="c28">R1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c31">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c28">R1</font></span><br/><div class="add"><a NAME="E2:73_1_5_2_5_2_1"/>
<font color="Maroon" title="c31">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_5_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:73_1_5_2_5_2_1_1"/>
not <font color="Maroon" title="c31">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:73_1_5_2_5_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 
<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/>
<span class="kw">hence </span><a NAME="E3:73_1_5_2_5_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2_5"><span class="lab"><font color="Green" title="E78">A49</font></span></a>, <a class="txt" href="#E1:73_1_5_2_5_2_1"><span class="lab"><font color="Green" title="E80">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:73_1_5_2_5_2_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c31">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c28">R1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2_5_2_1"><span class="lab"><font color="Green" title="E80">A54</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_5_2_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_5_2_2"/>
not <font color="Maroon" title="c28">R1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c31">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c28">R1</font></span><br/><div class="add"><a NAME="E2:73_1_5_2_5_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A55</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c32">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E3:73_1_5_2_5_2_2"/>
<font color="Maroon" title="c28">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c32">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_5_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">z</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="c33">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> or <font color="Maroon" title="c33">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c32">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_5_2_2_1"/><span class="lab"><font color="Green" title="E81">A56</font></span>: 
<font color="Maroon" title="c33">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</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="c33">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c32">f</font></span><br/>

<a NAME="E2:73_1_5_2_5_2_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c33">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E3:73_1_5_2_5_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A57</font></span>: 
<font color="Maroon" title="c33">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>;<br/>
<a NAME="E4:73_1_5_2_5_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A58</font></span>: 
<font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E1:73_1_5_2_5_2_2_1"><span class="lab"><font color="Green" title="E81">A56</font></span></a></span>;<br/>
<a NAME="E5:73_1_5_2_5_2_2_1"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ] &amp; <font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E1:73_1_5_2_5_2_2_1"><span class="lab"><font color="Green" title="E81">A56</font></span></a>, <a class="txt" href="#E3:73_1_5_2_5_2_2_1"><span class="lab"><font color="Green" title="E82">A57</font></span></a></span>;<br/>
<a NAME="E6:73_1_5_2_5_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A59</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E4:73_1_5_2_5_2_2_1"><span class="lab"><font color="Green" title="E83">A58</font></span></a></span>;<br/>
<a NAME="E7:73_1_5_2_5_2_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E8:73_1_5_2_5_2_2_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_5_2_2_1"><span class="lab"><font color="Green" title="E82">A57</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E9:73_1_5_2_5_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c33">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_5"><span class="lab"><font color="Green" title="E79">A53</font></span></a>, <a class="txt" href="#E6:73_1_5_2_5_2_2_1"><span class="lab"><font color="Green" title="E84">A59</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:73_1_5_2_5_2_2_1"/>
<font color="Maroon" title="c33">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c32">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_5_2_5_2_2"><span class="lab"><font color="Green" title="E80">A55</font></span></a>, <a class="txt" href="#E6:73_1_5_2_5_2_2_1"><span class="lab"><font color="Green" title="E84">A59</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1: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="E4:73_1_5_2_5_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A60</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c32">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">R1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_5_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">x1</font>, <font color="Maroon" title="c34">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c33">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> &amp; <font color="Maroon" title="c34">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> &amp; <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x2</font> implies <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73_1_5_2_5_2_2_2"/><span class="lab"><font color="Green" title="E82">A61</font></span>: 
<font color="Maroon" title="c33">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 <span class="kw">and </span><br/><a NAME="E2:73_1_5_2_5_2_2_2"/><span class="lab"><font color="Green" title="E83">A62</font></span>: 
<font color="Maroon" title="c34">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 <span class="kw">and </span><br/><a NAME="E3:73_1_5_2_5_2_2_2"/><span class="lab"><font color="Green" title="E84">A63</font></span>: 
<font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x2</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="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font></span><br/><a NAME="E4:73_1_5_2_5_2_2_2"/>
( <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c33">x1</font>) &amp; <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">x2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_5"><span class="lab"><font color="Green" title="E79">A53</font></span></a>, <a class="txt" href="#E1:73_1_5_2_5_2_2_2"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="txt" href="#E2:73_1_5_2_5_2_2_2"><span class="lab"><font color="Green" title="E83">A62</font></span></a></span>;<br/><a NAME="E5:73_1_5_2_5_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A64</font></span>: 
( <font color="Maroon" title="c33">x1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; <font color="Maroon" title="c33">x1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_5_2_2_2"><span class="lab"><font color="Green" title="E84">A63</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E6:73_1_5_2_5_2_2_2"/><span class="lab"><font color="Green" title="E86">A65</font></span>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c34">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_5_2_5_2_2_2"><span class="lab"><font color="Green" title="E83">A62</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E7:73_1_5_2_5_2_2_2"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2_5_2_2_2"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c33">x1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">x2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c34">x2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_5_2_2_2"><span class="lab"><font color="Green" title="E85">A64</font></span></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>
<br/>.= 
<font color="Maroon" title="c34">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_5_2_2_2"><span class="lab"><font color="Green" title="E86">A65</font></span></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>
;<br/> <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="E6:73_1_5_2_5_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_5_2_5_2_2"><span class="lab"><font color="Green" title="E80">A55</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E7:73_1_5_2_5_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">R</font>,<font color="Maroon" title="c28">R1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_5_2_5_2_2"><span class="lab"><font color="Green" title="E80">A55</font></span></a>, <a class="txt" href="#E4:73_1_5_2_5_2_2"><span class="lab"><font color="Green" title="E81">A60</font></span></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:73_1_5_2_5_2_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c31">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c28">R1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E19:73_1_5_2"/><span class="lab"><font color="Green" title="E78">A66</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="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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</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> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c10">Aa</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T8" target="_self" title="SIMPLEX1:th.8">Th8</a>, <a class="ref" href="simplex1.html#T10" target="_self" title="SIMPLEX1:th.10">Th10</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c32">DX</font> = <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> ;<br/><a NAME="E20:73_1_5_2"/><span class="lab"><font color="Green" title="E79">A67</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  <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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">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="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  or <font color="Maroon" title="c33">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_6"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</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="c33">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:73_1_5_2_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E3:73_1_5_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c33">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a></span>;<br/>
<a NAME="E4:73_1_5_2_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">x</font> &amp; <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:73_1_5_2_6"/>
<font color="Maroon" title="c33">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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="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><span class="kw">set </span><font color="Maroon" title="c33">RDX</font> = <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c34">DX</font> = <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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="c7">A</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="#E20:73_1_5_2"><span class="lab"><font color="Green" title="E79">A67</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">reconsider </span><font color="Maroon" title="c35">DX</font> = <font color="Maroon" title="c34">DX</font> as   <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="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="c7">A</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="#E20:73_1_5_2"><span class="lab"><font color="Green" title="E79">A67</font></span></a>, <a class="ref" href="simplex0.html#T14" title="SIMPLEX0:th.14">SIMPLEX0:14</a></span>;<br/><a NAME="E23:73_1_5_2"/><span class="lab"><font color="Green" title="E80">A68</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c35">DX</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:73_1_5_2_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c35">DX</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">xy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_7"/><span class="lab"><font color="Green" title="E81">A69</font></span>: 
<font color="Maroon" title="c36">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c35">DX</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/>
<span class="kw">consider </span><font color="Maroon" title="c37">x</font>, <font color="Maroon" title="c38">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:73_1_5_2_7"/><span class="lab"><font color="Green" title="E82">A70</font></span>: 
<font color="Maroon" title="c36">xy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c37">x</font>,<font color="Maroon" title="c38">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_7"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E6:73_1_5_2_7"/><span class="lab"><font color="Green" title="E83">A71</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c35">DX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_7"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="txt" href="#E5:73_1_5_2_7"><span class="lab"><font color="Green" title="E82">A70</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E7:73_1_5_2_7"/><span class="kw">then </span>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">DX</font> &amp; <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_5_2_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_7"><span class="lab"><font color="Green" title="E83">A71</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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><span class="lab"><font color="Green" title="E81">A72</font></span>: <font color="Maroon" title="c20">m1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</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="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c6">m</font>

;<br/><span class="kw">set </span><font color="Maroon" title="c36">FA</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E25:73_1_5_2"/><span class="lab"><font color="Green" title="E82">A73</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c10">Aa</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="#E26:73_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E22:73_1"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="simplex0.html#T31" title="SIMPLEX0:th.31">SIMPLEX0:31</a></span>;<br/><a NAME="E26:73_1_5_2"/><span class="lab"><font color="Green" title="E83">A74</font></span>: 
not  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E10:73_1"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E12:73_1"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E1:73_1_5_2"><span class="lab"><font color="Green" title="E68">A30</font></span></a></span>;<br/><a NAME="E27:73_1_5_2"/><span class="lab"><font color="Green" title="E84">A75</font></span>: 
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="c10">Aa</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="c10">Aa</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/><span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">v</font> be   <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>; <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">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="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Aa</font> &amp; <font color="Maroon" title="c37">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="b1">B</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c38">B</font> be   <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"> ( <font color="Maroon" title="c38">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Aa</font> &amp; <font color="Maroon" title="c37">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="Maroon" title="c38">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">B</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:73_1_5_2_9"/><span class="lab"><font color="Green" title="E85">A76</font></span>: 
( <font color="Maroon" title="c38">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Aa</font> &amp; <font color="Maroon" title="c37">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="Maroon" title="c38">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">B</font></span><br/>

<a NAME="E2:73_1_5_2_9"/>
<font color="Maroon" title="c37">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c10">Aa</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="#E26:73_1_5_2"><span class="lab"><font color="Green" title="E83">A74</font></span></a></span>;<br/>
<a NAME="E3:73_1_5_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E18:73_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E23:73_1"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="txt" href="#E1:73_1_5_2_9"><span class="lab"><font color="Green" title="E85">A76</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="E4:73_1_5_2_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:73_1_5_2"><span class="lab"><font color="Green" title="E82">A73</font></span></a>, <a class="txt" href="#E26:73_1_5_2"><span class="lab"><font color="Green" title="E83">A74</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</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="E28:73_1_5_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Aa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">y</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="c37">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c37">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_10"/>
<font color="Maroon" title="c37">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></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="c37">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_10"/><span class="lab"><font color="Green" title="E85">A77</font></span>: 
<font color="Maroon" title="c38">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_10"/><span class="lab"><font color="Green" title="E86">A78</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p3">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> <font color="Maroon" title="c1">k</font>,<span class="p4">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c39">v</font> = <font color="Maroon" title="c38">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c10">Aa</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="#E25:73_1_5_2"><span class="lab"><font color="Green" title="E82">A73</font></span></a>, <a class="txt" href="#E3:73_1_5_2_10"><span class="lab"><font color="Green" title="E85">A77</font></span></a></span>;<br/>
<a NAME="E6:73_1_5_2_10"/>
<font color="Maroon" title="c39">v</font> is  <a href="simplex0.html#V1" title="SIMPLEX0:attr.1">vertex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:73_1_5_2"><span class="lab"><font color="Green" title="E82">A73</font></span></a>, <a class="txt" href="#E3:73_1_5_2_10"><span class="lab"><font color="Green" title="E85">A77</font></span></a>, <a class="ref" href="simplex0.html#D4" title="SIMPLEX0:def.4">SIMPLEX0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c40">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:73_1_5_2_10"/><span class="lab"><font color="Green" title="E87">A79</font></span>: 
not <font color="Maroon" title="c40">S</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">and </span><br/><a NAME="E9:73_1_5_2_10"/><span class="lab"><font color="Green" title="E88">A80</font></span>: 
<font color="Maroon" title="c39">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D3" title="SIMPLEX0:def.3">SIMPLEX0:def 3</a></span>;<br/>
<a NAME="E10:73_1_5_2_10"/><span class="lab"><font color="Green" title="E89">A81</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c40">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_5_2_10"><span class="lab"><font color="Green" title="E87">A79</font></span></a>, <a class="ref" href="simplex1.html#T5" target="_self" title="SIMPLEX1:th.5">Th5</a></span>;<br/>
<a NAME="E11:73_1_5_2_10"/>
<font color="Maroon" title="c40">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c40">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T2" title="RLAFFIN1:th.2">RLAFFIN1:2</a></span>;<br/>
<a NAME="E12:73_1_5_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A82</font></span>: 
<font color="Maroon" title="c39">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c40">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2_10"><span class="lab"><font color="Green" title="E88">A80</font></span></a></span>;<br/>
<a NAME="E13:73_1_5_2_10"/>
<font color="Maroon" title="c38">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <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="c10">Aa</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="#E26:73_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E22:73_1"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="txt" href="#E3:73_1_5_2_10"><span class="lab"><font color="Green" title="E85">A77</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="simplex0.html#T31" title="SIMPLEX0:th.31">SIMPLEX0:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:73_1_5_2_10"/>
<font color="Maroon" title="c37">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:73_1_5_2"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="txt" href="#E27:73_1_5_2"><span class="lab"><font color="Green" title="E84">A75</font></span></a>, <a class="txt" href="#E4:73_1_5_2_10"><span class="lab"><font color="Green" title="E86">A78</font></span></a>, <a class="txt" href="#E10:73_1_5_2_10"><span class="lab"><font color="Green" title="E89">A81</font></span></a>, <a class="txt" href="#E12:73_1_5_2_10"><span class="lab"><font color="Green" title="E90">A82</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c37">FA</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> as   <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="c10">Aa</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="c10">Aa</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:73_1_5_2"><span class="lab"><font color="Green" title="E82">A73</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c38">XXa</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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c39">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E31:73_1_5_2"/><span class="lab"><font color="Green" title="E85">A83</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</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="Maroon" title="c39">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="txt" href="#E1:73_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E27:73_1_5_2"><span class="lab"><font color="Green" title="E84">A75</font></span></a></span>;<br/><a NAME="E32:73_1_5_2"/><span class="lab"><font color="Green" title="E86">A84</font></span>: 
( <font color="Maroon" title="c6">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <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="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a>, <a class="ref" href="xreal_1.html#T33" title="XREAL_1:th.33">XREAL_1:33</a></span>;<br/><a NAME="E33:73_1_5_2"/><span class="lab"><font color="Green" title="E87">A85</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c31">R</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c40">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c31">R</font>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_11"/>
<font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c31">R</font>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="E3:73_1_5_2_11"/><span class="lab"><font color="Green" title="E88">A86</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_11"/><span class="lab"><font color="Green" title="E89">A87</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Aa</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_5_2_11"/><span class="lab"><font color="Green" title="E90">A88</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">XX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> ;<br/>
<a NAME="E6:73_1_5_2_11"/><span class="lab"><font color="Green" title="E91">A89</font></span>: 
<font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c43">w</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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_11_1"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_11_1"/><span class="lab"><font color="Green" title="E92">A90</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">s</font>,<font color="Maroon" title="c43">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_11_1"/><span class="lab"><font color="Green" title="E93">A91</font></span>: 
<font color="Maroon" title="c44">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/>
<a NAME="E5:73_1_5_2_11_1"/>
<font color="Maroon" title="c43">w</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E3:73_1_5_2_11_1"><span class="lab"><font color="Green" title="E92">A90</font></span></a></span>;<br/>
<a NAME="E6:73_1_5_2_11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A92</font></span>: 
 ex <font color="Olive" title="b1">W</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> st <font color="Maroon" title="c43">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font>
 
;<br/>
<a NAME="E7:73_1_5_2_11_1"/>
( <font color="Maroon" title="c44">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font> &amp; <font color="Maroon" title="c44">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c43">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E3:73_1_5_2_11_1"><span class="lab"><font color="Green" title="E92">A90</font></span></a>, <a class="txt" href="#E4:73_1_5_2_11_1"><span class="lab"><font color="Green" title="E93">A91</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_5_2_11_1"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_11_1"><span class="lab"><font color="Green" title="E94">A92</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="E7:73_1_5_2_11"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_11_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c43">w</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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  or <font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_11_2"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">W</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</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="E3:73_1_5_2_11_2"/><span class="lab"><font color="Green" title="E92">A93</font></span>: 
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">W</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_11_2"/><span class="lab"><font color="Green" title="E93">A94</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c44">W</font>
 ;<br/>
<a NAME="E5:73_1_5_2_11_2"/>
( <font color="Maroon" title="c44">W</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon" title="c41">S</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_11"><span class="lab"><font color="Green" title="E89">A87</font></span></a></span>;<br/>
<a NAME="E6:73_1_5_2_11_2"/><span class="kw">then </span>
( <font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c44">W</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E4:73_1_5_2_11_2"><span class="lab"><font color="Green" title="E93">A94</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:73_1_5_2_11_2"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_11_2"><span class="lab"><font color="Green" title="E92">A93</font></span></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</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="E8:73_1_5_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A95</font></span>: 
<font color="Maroon" title="c31">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_11"><span class="lab"><font color="Green" title="E91">A89</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E9:73_1_5_2_11"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E5:73_1_5_2_11"><span class="lab"><font color="Green" title="E90">A88</font></span></a>, <a class="ref" href="simplex1.html#T45" target="_self" title="SIMPLEX1:th.45">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:73_1_5_2_11"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c31">R</font>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_11"><span class="lab"><font color="Green" title="E88">A86</font></span></a>, <a class="txt" href="#E8:73_1_5_2_11"><span class="lab"><font color="Green" title="E92">A95</font></span></a>, <a class="ref" href="relat_1.html#D16" title="RELAT_1:def.16">RELAT_1:def 16</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="E34:73_1_5_2"/><span class="lab"><font color="Green" title="E88">A96</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#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="c7">A</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="#E17:73_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="ref" href="simplex1.html#T32" target="_self" title="SIMPLEX1:th.32">Th32</a></span>;<br/><a NAME="E35:73_1_5_2"/><span class="lab"><font color="Green" title="E89">A97</font></span>: 
<font color="Maroon" title="c21">M</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <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="E36:73_1_5_2"/><span class="lab"><font color="Green" title="E90">A98</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">YA</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c28">R1</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c40">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">YA</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c28">R1</font>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_12"/>
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">YA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c28">R1</font>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</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="E3:73_1_5_2_12"/><span class="lab"><font color="Green" title="E91">A99</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_12"/><span class="lab"><font color="Green" title="E92">A100</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">FS</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font>;<br/>
<a NAME="E5:73_1_5_2_12"/><span class="lab"><font color="Green" title="E93">A101</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_12"><span class="lab"><font color="Green" title="E92">A100</font></span></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/>
<a NAME="E6:73_1_5_2_12"/><span class="lab"><font color="Green" title="E94">A102</font></span>: 
not <font color="Maroon" title="c10">Aa</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E1:73_1_5_2"><span class="lab"><font color="Green" title="E68">A30</font></span></a></span>;<br/>
<a NAME="E7:73_1_5_2_12"/><span class="lab"><font color="Green" title="E95">A103</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_12"><span class="lab"><font color="Green" title="E91">A99</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:73_1_5_2_12"/><span class="lab"><font color="Green" title="E96">A104</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:73_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="simplex0.html#T17" title="SIMPLEX0:th.17">SIMPLEX0:17</a></span>;<br/>
<a NAME="E9:73_1_5_2_12"/><span class="lab"><font color="Green" title="E97">A105</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <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="#E34:73_1_5_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a>, <a class="txt" href="#E15:73_1_5_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="txt" href="#E35:73_1_5_2"><span class="lab"><font color="Green" title="E89">A97</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c43">FS</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c10">Aa</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_12"><span class="lab"><font color="Green" title="E93">A101</font></span></a>, <a class="txt" href="#E8:73_1_5_2_12"><span class="lab"><font color="Green" title="E96">A104</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<a NAME="E11:73_1_5_2_12"/>
<font color="Maroon" title="c43">FS</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_12"><span class="lab"><font color="Green" title="E93">A101</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:73_1_5_2_12"/><span class="lab"><font color="Green" title="E98">A106</font></span>: 
<font color="Maroon" title="c44">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font>
 <span class="kw">and </span><br/><a NAME="E14:73_1_5_2_12"/><span class="lab"><font color="Green" title="E99">A107</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">and </span><br/><a NAME="E15:73_1_5_2_12"/><span class="lab"><font color="Green" title="E100">A108</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">b</font> holds <br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E6:73_1_5_2_12"><span class="lab"><font color="Green" title="E94">A102</font></span></a>, <a class="txt" href="#E9:73_1_5_2_12"><span class="lab"><font color="Green" title="E97">A105</font></span></a>, <a class="ref" href="simplex1.html#T2" target="_self" title="SIMPLEX1:th.2">Th2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c45">a1</font>, <font color="Maroon" title="c46">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:73_1_5_2_12"/><span class="lab"><font color="Green" title="E101">A109</font></span>: 
<font color="Maroon" title="c45">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c46">a2</font>
 <span class="kw">and </span><br/><a NAME="E18:73_1_5_2_12"/><span class="lab"><font color="Green" title="E102">A110</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:73_1_5_2_12"><span class="lab"><font color="Green" title="E99">A107</font></span></a>, <a class="ref" href="card_2.html#T79" title="CARD_2:th.79">CARD_2:79</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c47">S1</font> = <font color="Maroon" title="c41">S</font> <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"><font color="Maroon" title="c45">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <font color="Maroon" title="c48">S2</font> = <font color="Maroon" title="c41">S</font> <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"><font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> ;<br/>
<a NAME="E20:73_1_5_2_12"/><span class="lab"><font color="Green" title="E103">A111</font></span>: 
<font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E21:73_1_5_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A112</font></span>: 
<font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c48">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2_12"><span class="lab"><font color="Green" title="E101">A109</font></span></a>, <a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E22:73_1_5_2_12"/><span class="lab"><font color="Green" title="E105">A114</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c47">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2_12"><span class="lab"><font color="Green" title="E97">A105</font></span></a>, <a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E20:73_1_5_2_12"><span class="lab"><font color="Green" title="E103">A111</font></span></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></span>;<br/>
<a NAME="E23:73_1_5_2_12"/><span class="lab"><font color="Green" title="E106">A115</font></span>: 
<font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E24:73_1_5_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A116</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c48">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2_12"><span class="lab"><font color="Green" title="E97">A105</font></span></a>, <a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c49">S1</font> = <font color="Maroon" title="c47">S1</font>, <font color="Maroon" title="c50">S2</font> = <font color="Maroon" title="c48">S2</font> as    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="#E34:73_1_5_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a>, <a class="txt" href="#E32:73_1_5_2"><span class="lab"><font color="Green" title="E86">A84</font></span></a>, <a class="txt" href="#E24:73_1_5_2"><span class="lab"><font color="Green" title="E81">A72</font></span></a>, <a class="txt" href="#E15:73_1_5_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="txt" href="#E22:73_1_5_2_12"><span class="lab"><font color="Green" title="E105">A114</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/>
<a NAME="E26:73_1_5_2_12"/><span class="lab"><font color="Green" title="E108">A117</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c45">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E20:73_1_5_2_12"><span class="lab"><font color="Green" title="E103">A111</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E27:73_1_5_2_12"/><span class="lab"><font color="Green" title="E109">A118</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_5_2_12"><span class="lab"><font color="Green" title="E96">A104</font></span></a>, <a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E23:73_1_5_2_12"><span class="lab"><font color="Green" title="E106">A115</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E28:73_1_5_2_12"/><span class="lab"><font color="Green" title="E110">A119</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E23:73_1_5_2_12"><span class="lab"><font color="Green" title="E106">A115</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E29:73_1_5_2_12"/><span class="lab"><font color="Green" title="E111">A120</font></span>: 
<font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c51">Y</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="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_12_1"/>
<font color="Maroon" title="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c52">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E112">A121</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c52">X</font>,<font color="Maroon" title="c51">Y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E113">A122</font></span>: 
<font color="Maroon" title="c52">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/>
<a NAME="E5:73_1_5_2_12_1"/>
<font color="Maroon" title="c51">Y</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E112">A121</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c53">W</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="E7:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E114">A123</font></span>: 
<font color="Maroon" title="c51">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">W</font>
 <span class="kw">and </span><br/><a NAME="E8:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E115">A124</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c53">W</font>
 ;<br/>
<a NAME="E9:73_1_5_2_12_1"/>
<font color="Maroon" title="c52">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E113">A122</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:73_1_5_2_12_1"/><span class="kw">then </span>
<font color="Maroon" title="c53">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_12"><span class="lab"><font color="Green" title="E91">A99</font></span></a>, <a class="txt" href="#E3:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E112">A121</font></span></a>, <a class="txt" href="#E7:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E114">A123</font></span></a></span>;<br/>
<a NAME="E11:73_1_5_2_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A125</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c53">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E115">A124</font></span></a>, <a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c54">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E117">A126</font></span>: 
<font color="Maroon" title="c54">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c43">FS</font>
 <span class="kw">and </span><br/><a NAME="E14:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E118">A127</font></span>: 
<font color="Maroon" title="c54">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c53">W</font>
 <span class="kw">and </span><br/><a NAME="E15:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E119">A128</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1_5_2_12"><span class="lab"><font color="Green" title="E98">A106</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E16:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E120">A129</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c54">w</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c53">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E118">A127</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E17:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E121">A130</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c53">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_12_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c55">s</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="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c53">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_12_1_1"/><span class="lab"><font color="Green" title="E122">A131</font></span>: 
<font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></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="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c53">W</font></span><br/>

<a NAME="E2:73_1_5_2_12_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A132</font></span>: 
<font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c43">FS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_5_2_12"><span class="lab"><font color="Green" title="E96">A104</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:73_1_5_2_12_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A133</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_12"><span class="lab"><font color="Green" title="E93">A101</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c56">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:73_1_5_2_12_1_1"/><span class="lab"><font color="Green" title="E125">A134</font></span>: 
<font color="Maroon" title="c56">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c43">FS</font>
 <span class="kw">and </span><br/><a NAME="E6:73_1_5_2_12_1_1"/><span class="lab"><font color="Green" title="E126">A135</font></span>: 
<font color="Maroon" title="c56">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c53">W</font>
 <span class="kw">and </span><br/><a NAME="E7:73_1_5_2_12_1_1"/><span class="lab"><font color="Green" title="E127">A136</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c56">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E116">A125</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E8:73_1_5_2_12_1_1"/>
not <font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E1:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E122">A131</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:73_1_5_2_12_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E123">A132</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E10:73_1_5_2_12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E11:73_1_5_2_12_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:73_1_5_2_12"><span class="lab"><font color="Green" title="E100">A108</font></span></a>, <a class="txt" href="#E3:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E124">A133</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c57">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:73_1_5_2_12_1_1"/><span class="lab"><font color="Green" title="E128">A137</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c57">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></span>;<br/>
<a NAME="E14:73_1_5_2_12_1_1"/><span class="lab"><font color="Green" title="E129">A138</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:73_1_5_2_12_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A139</font></span>: 
<font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E123">A132</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E16:73_1_5_2_12_1_1"/>
<font color="Maroon" title="c56">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E125">A134</font></span></a>, <a class="txt" href="#E7:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E127">A136</font></span></a>, <a class="txt" href="#E14:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E129">A138</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E17:73_1_5_2_12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c56">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c57">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E128">A137</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:73_1_5_2_12_1_1"/>
<font color="Maroon" title="c55">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c53">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E126">A135</font></span></a>, <a class="txt" href="#E13:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E128">A137</font></span></a>, <a class="txt" href="#E15:73_1_5_2_12_1_1"><span class="lab"><font color="Green" title="E130">A139</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E18:73_1_5_2_12_1"/>
<font color="Maroon" title="c44">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E19:73_1_5_2_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A140</font></span>: 
<font color="Maroon" title="c54">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E117">A126</font></span></a>, <a class="txt" href="#E15:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E119">A128</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E20:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E123">A141</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c53">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:73_1_5_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a>, <a class="txt" href="#E32:73_1_5_2"><span class="lab"><font color="Green" title="E86">A84</font></span></a>, <a class="txt" href="#E24:73_1_5_2"><span class="lab"><font color="Green" title="E81">A72</font></span></a>, <a class="txt" href="#E15:73_1_5_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/>
<a NAME="E21:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E124">A142</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c45">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c45">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:73_1_5_2_12"><span class="lab"><font color="Green" title="E108">A117</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E22:73_1_5_2_12_1"/><span class="lab"><font color="Green" title="E125">A143</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:73_1_5_2_12"><span class="lab"><font color="Green" title="E110">A119</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_5_2_12_1_2"/>
( <font color="Maroon" title="c54">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">a1</font> or <font color="Maroon" title="c54">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">a2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E19:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E122">A140</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_12_1_2_1"/>
<font color="Maroon" title="c54">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">a1</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="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c54">w</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c45">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E124">A142</font></span></a>, <a class="ref" href="xboole_1.html#T52" title="XBOOLE_1:th.52">XBOOLE_1:52</a></span>
<br/>.= 
<font color="Maroon" title="c50">S2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2_12"><span class="lab"><font color="Green" title="E101">A109</font></span></a>, <a class="ref" href="zfmisc_1.html#T23" title="ZFMISC_1:th.23">ZFMISC_1:23</a></span>
;<br/><a NAME="E3:73_1_5_2_12_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c50">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:73_1_5_2_12"><span class="lab"><font color="Green" title="E107">A116</font></span></a>, <a class="txt" href="#E16:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E120">A129</font></span></a>, <a class="txt" href="#E17:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E121">A130</font></span></a>, <a class="txt" href="#E20:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E123">A141</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:73_1_5_2_12_1_2_1"/>
<font color="Maroon" title="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E114">A123</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_12_1_2_2"/>
<font color="Maroon" title="c54">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">a2</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="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c54">w</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c45">a1</font>,<font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E125">A143</font></span></a>, <a class="ref" href="xboole_1.html#T52" title="XBOOLE_1:th.52">XBOOLE_1:52</a></span>
<br/>.= 
<font color="Maroon" title="c49">S1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2_12"><span class="lab"><font color="Green" title="E101">A109</font></span></a>, <a class="ref" href="zfmisc_1.html#T23" title="ZFMISC_1:th.23">ZFMISC_1:23</a></span>
;<br/><a NAME="E3:73_1_5_2_12_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c49">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:73_1_5_2_12"><span class="lab"><font color="Green" title="E105">A114</font></span></a>, <a class="txt" href="#E16:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E120">A129</font></span></a>, <a class="txt" href="#E17:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E121">A130</font></span></a>, <a class="txt" href="#E20:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E123">A141</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:73_1_5_2_12_1_2_2"/>
<font color="Maroon" title="c51">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_5_2_12_1"><span class="lab"><font color="Green" title="E114">A123</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E30:73_1_5_2_12"/><span class="lab"><font color="Green" title="E112">A144</font></span>: 
<font color="Maroon" title="c41">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> <font color="Maroon" title="c14">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:73_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="simplex0.html#T17" title="SIMPLEX0:th.17">SIMPLEX0:17</a></span>;<br/>
<a NAME="E31:73_1_5_2_12"/><span class="lab"><font color="Green" title="E113">A145</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_5_2_12"><span class="lab"><font color="Green" title="E96">A104</font></span></a>, <a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E20:73_1_5_2_12"><span class="lab"><font color="Green" title="E103">A111</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E32:73_1_5_2_12"/><span class="lab"><font color="Green" title="E114">A146</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E20:73_1_5_2_12"><span class="lab"><font color="Green" title="E103">A111</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E33:73_1_5_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A147</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E34:73_1_5_2_12"/><span class="lab"><font color="Green" title="E116">A148</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E23:73_1_5_2_12"><span class="lab"><font color="Green" title="E106">A115</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E35:73_1_5_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A149</font></span>: 
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E36:73_1_5_2_12"/><span class="lab"><font color="Green" title="E118">A150</font></span>: 
( <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">S</font> &amp; <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c49">S1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2_12"><span class="lab"><font color="Green" title="E101">A109</font></span></a>, <a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E23:73_1_5_2_12"><span class="lab"><font color="Green" title="E106">A115</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E37:73_1_5_2_12"/><span class="lab"><font color="Green" title="E119">A151</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_12_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c51">z</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font> or <font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_12_2"/><span class="lab"><font color="Green" title="E120">A152</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_5_2_12_2_1"/>
( <font color="Maroon" title="c51">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">b</font> or <font color="Maroon" title="c51">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_12_2_1_1"/><span class="lab"><font color="Green" title="E121">A153</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">b</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font></span><br/><div class="add"><a NAME="E2:73_1_5_2_12_2_1_1"/>
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:73_1_5_2_12"><span class="lab"><font color="Green" title="E112">A144</font></span></a>, <a class="txt" href="#E27:73_1_5_2_12"><span class="lab"><font color="Green" title="E109">A118</font></span></a>, <a class="txt" href="#E36:73_1_5_2_12"><span class="lab"><font color="Green" title="E118">A150</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:73_1_5_2_12_2_1_1"/>
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:73_1_5_2_12"><span class="lab"><font color="Green" title="E116">A148</font></span></a>, <a class="txt" href="#E1:73_1_5_2_12_2_1_1"><span class="lab"><font color="Green" title="E121">A153</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_12_2_1_2"/><span class="lab"><font color="Green" title="E121">A154</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">b</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c52">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_12_2_1_2"/><span class="lab"><font color="Green" title="E122">A155</font></span>: 
<font color="Maroon" title="c52">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_12_2_1_2"/><span class="lab"><font color="Green" title="E123">A156</font></span>: 
<font color="Maroon" title="c52">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_5_2_12_2_1_2"/><span class="lab"><font color="Green" title="E124">A157</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c52">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_12"><span class="lab"><font color="Green" title="E92">A100</font></span></a>, <a class="txt" href="#E1:73_1_5_2_12_2"><span class="lab"><font color="Green" title="E120">A152</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><a NAME="E6:73_1_5_2_12_2_1_2"/>
<font color="Maroon" title="c52">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c49">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:73_1_5_2_12"><span class="lab"><font color="Green" title="E113">A145</font></span></a>, <a class="txt" href="#E33:73_1_5_2_12"><span class="lab"><font color="Green" title="E115">A147</font></span></a>, <a class="txt" href="#E1:73_1_5_2_12_2_1_2"><span class="lab"><font color="Green" title="E121">A154</font></span></a>, <a class="txt" href="#E4:73_1_5_2_12_2_1_2"><span class="lab"><font color="Green" title="E123">A156</font></span></a>, <a class="txt" href="#E5:73_1_5_2_12_2_1_2"><span class="lab"><font color="Green" title="E124">A157</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E7:73_1_5_2_12_2_1_2"/>
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_12_2_1_2"><span class="lab"><font color="Green" title="E122">A155</font></span></a>, <a class="txt" href="#E5:73_1_5_2_12_2_1_2"><span class="lab"><font color="Green" title="E124">A157</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E38:73_1_5_2_12"/><span class="lab"><font color="Green" title="E120">A158</font></span>: 
<font color="Maroon" title="c41">S</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> 
 
;<br/>
<a NAME="E39:73_1_5_2_12"/><span class="lab"><font color="Green" title="E121">A159</font></span>: 
( <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">S</font> &amp; <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c50">S2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2_12"><span class="lab"><font color="Green" title="E101">A109</font></span></a>, <a class="txt" href="#E18:73_1_5_2_12"><span class="lab"><font color="Green" title="E102">A110</font></span></a>, <a class="txt" href="#E20:73_1_5_2_12"><span class="lab"><font color="Green" title="E103">A111</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E40:73_1_5_2_12"/><span class="lab"><font color="Green" title="E122">A160</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_12_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c51">z</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font> or <font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_12_3"/><span class="lab"><font color="Green" title="E123">A161</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_5_2_12_3_1"/>
( <font color="Maroon" title="c51">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">b</font> or <font color="Maroon" title="c51">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_12_3_1_1"/><span class="lab"><font color="Green" title="E124">A162</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">b</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font></span><br/><div class="add"><a NAME="E2:73_1_5_2_12_3_1_1"/>
<font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:73_1_5_2_12"><span class="lab"><font color="Green" title="E113">A145</font></span></a>, <a class="txt" href="#E30:73_1_5_2_12"><span class="lab"><font color="Green" title="E112">A144</font></span></a>, <a class="txt" href="#E39:73_1_5_2_12"><span class="lab"><font color="Green" title="E121">A159</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:73_1_5_2_12_3_1_1"/>
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:73_1_5_2_12"><span class="lab"><font color="Green" title="E114">A146</font></span></a>, <a class="txt" href="#E1:73_1_5_2_12_3_1_1"><span class="lab"><font color="Green" title="E124">A162</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_5_2_12_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_5_2_12_3_1_2"/><span class="lab"><font color="Green" title="E124">A163</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">b</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="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c52">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_12_3_1_2"/><span class="lab"><font color="Green" title="E125">A164</font></span>: 
<font color="Maroon" title="c52">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_12_3_1_2"/><span class="lab"><font color="Green" title="E126">A165</font></span>: 
<font color="Maroon" title="c52">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_5_2_12_3_1_2"/><span class="lab"><font color="Green" title="E127">A166</font></span>: 
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c52">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_12"><span class="lab"><font color="Green" title="E92">A100</font></span></a>, <a class="txt" href="#E1:73_1_5_2_12_3"><span class="lab"><font color="Green" title="E123">A161</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><a NAME="E6:73_1_5_2_12_3_1_2"/>
<font color="Maroon" title="c52">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c50">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:73_1_5_2_12"><span class="lab"><font color="Green" title="E109">A118</font></span></a>, <a class="txt" href="#E35:73_1_5_2_12"><span class="lab"><font color="Green" title="E117">A149</font></span></a>, <a class="txt" href="#E1:73_1_5_2_12_3_1_2"><span class="lab"><font color="Green" title="E124">A163</font></span></a>, <a class="txt" href="#E4:73_1_5_2_12_3_1_2"><span class="lab"><font color="Green" title="E126">A165</font></span></a>, <a class="txt" href="#E5:73_1_5_2_12_3_1_2"><span class="lab"><font color="Green" title="E127">A166</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E7:73_1_5_2_12_3_1_2"/>
<font color="Maroon" title="c51">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_12_3_1_2"><span class="lab"><font color="Green" title="E125">A164</font></span></a>, <a class="txt" href="#E5:73_1_5_2_12_3_1_2"><span class="lab"><font color="Green" title="E127">A166</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E41:73_1_5_2_12"/>
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_12"><span class="lab"><font color="Green" title="E92">A100</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E42:73_1_5_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c49">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:73_1_5_2_12"><span class="lab"><font color="Green" title="E119">A151</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E43:73_1_5_2_12"/><span class="kw">then </span>
( <font color="Maroon" title="c41">S</font> <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"><font color="Maroon" title="c45">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">S</font> &amp; <font color="Maroon" title="c49">S1</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span>  )
 
<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="E44:73_1_5_2_12"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c49">S1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E38:73_1_5_2_12"><span class="lab"><font color="Green" title="E120">A158</font></span></a></span>;<br/>
<a NAME="E45:73_1_5_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A167</font></span>: 
<font color="Maroon" title="c49">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_5_2_12"><span class="lab"><font color="Green" title="E95">A103</font></span></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/>
<a NAME="E46:73_1_5_2_12"/>
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_12"><span class="lab"><font color="Green" title="E92">A100</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E47:73_1_5_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c50">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:73_1_5_2_12"><span class="lab"><font color="Green" title="E122">A160</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E48:73_1_5_2_12"/><span class="kw">then </span>
( <font color="Maroon" title="c41">S</font> <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"><font color="Maroon" title="c46">a2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">S</font> &amp; <font color="Maroon" title="c50">S2</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span>  )
 
<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="E49:73_1_5_2_12"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E38:73_1_5_2_12"><span class="lab"><font color="Green" title="E120">A158</font></span></a></span>;<br/>
<a NAME="E50:73_1_5_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c50">S2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_5_2_12"><span class="lab"><font color="Green" title="E95">A103</font></span></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/>
<a NAME="E51:73_1_5_2_12"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:73_1_5_2_12"><span class="lab"><font color="Green" title="E123">A167</font></span></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></span>;<br/>
<a NAME="E52:73_1_5_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A168</font></span>: 
<font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c49">S1</font>,<font color="Maroon" title="c50">S2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:73_1_5_2_12"><span class="lab"><font color="Green" title="E111">A120</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E53:73_1_5_2_12"/>
<font color="Maroon" title="c49">S1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c50">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:73_1_5_2_12"><span class="lab"><font color="Green" title="E104">A112</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E54:73_1_5_2_12"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:73_1_5_2_12"><span class="lab"><font color="Green" title="E124">A168</font></span></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E55:73_1_5_2_12"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c28">R1</font>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D16" title="RELAT_1:def.16">RELAT_1:def 16</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="E37:73_1_5_2"/><span class="lab"><font color="Green" title="E91">A169</font></span>: 
<font color="Maroon" title="c21">M</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <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/><a NAME="E38:73_1_5_2"/>
<font color="Maroon" title="c17">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c40">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="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</font> or <font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_13"/>
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</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="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</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="E3:73_1_5_2_13"/><span class="lab"><font color="Green" title="E92">A170</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_13"/><span class="lab"><font color="Green" title="E93">A171</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">FS</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font>;<br/>
<a NAME="E5:73_1_5_2_13"/><span class="lab"><font color="Green" title="E94">A172</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_13"><span class="lab"><font color="Green" title="E93">A171</font></span></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/>
<a NAME="E6:73_1_5_2_13"/><span class="lab"><font color="Green" title="E95">A173</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</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="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E34:73_1_5_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a>, <a class="txt" href="#E15:73_1_5_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="txt" href="#E35:73_1_5_2"><span class="lab"><font color="Green" title="E89">A97</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/>
<a NAME="E7:73_1_5_2_13"/><span class="lab"><font color="Green" title="E96">A174</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:73_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="simplex0.html#T17" title="SIMPLEX0:th.17">SIMPLEX0:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c43">FS</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c7">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_13"><span class="lab"><font color="Green" title="E94">A172</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c44">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:73_1_5_2_13"/><span class="lab"><font color="Green" title="E97">A175</font></span>: 
( <font color="Maroon" title="c44">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c43">FS</font> &amp; <font color="Maroon" title="c43">FS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E5:73_1_5_2_13"><span class="lab"><font color="Green" title="E94">A172</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c45">Ss</font> = <font color="Maroon" title="c41">S</font> <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"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E11:73_1_5_2_13"/>
<font color="Maroon" title="c43">FS</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_13"><span class="lab"><font color="Green" title="E94">A172</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E12:73_1_5_2_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A176</font></span>: 
<font color="Maroon" title="c43">FS</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_13"><span class="lab"><font color="Green" title="E95">A173</font></span></a>, <a class="ref" href="stirl2_1.html#T70" title="STIRL2_1:th.70">STIRL2_1:70</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E99">A177</font></span>: <font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T123" title="FUNCT_1:th.123">FUNCT_1:123</a></span>
<br/>.= 
<font color="Maroon" title="c7">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_13"><span class="lab"><font color="Green" title="E94">A172</font></span></a>, <a class="txt" href="#E7:73_1_5_2_13"><span class="lab"><font color="Green" title="E96">A174</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>
<br/>.= 
<font color="Maroon" title="c7">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c43">FS</font>,<font color="Maroon" title="c44">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D16" title="RELAT_1:def.16">RELAT_1:def 16</a></span>
<br/>.= 
<font color="Maroon" title="c10">Aa</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1_5_2_13"><span class="lab"><font color="Green" title="E97">A175</font></span></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></span>
;<br/>
<a NAME="E14:73_1_5_2_13"/>
<font color="Maroon" title="c41">S</font> <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"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_5_2_13"><span class="lab"><font color="Green" title="E96">A174</font></span></a>, <a class="txt" href="#E12:73_1_5_2_13"><span class="lab"><font color="Green" title="E98">A176</font></span></a>, <a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E15:73_1_5_2_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A178</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E13:73_1_5_2_13"><span class="lab"><font color="Green" title="E99">A177</font></span></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c46">Ss</font> = <font color="Maroon" title="c41">S</font> <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"><font color="Maroon" title="c44">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c47">Ss</font> = <font color="Maroon" title="c46">Ss</font> as    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="#E37:73_1_5_2"><span class="lab"><font color="Green" title="E91">A169</font></span></a>, <a class="txt" href="#E15:73_1_5_2_13"><span class="lab"><font color="Green" title="E100">A178</font></span></a>, <a class="ref" href="simplex0.html#T48" title="SIMPLEX0:th.48">SIMPLEX0:48</a></span>;<br/>
<a NAME="E18:73_1_5_2_13"/>
<font color="Maroon" title="c43">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c47">Ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c47">Ss</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>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E19:73_1_5_2_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A179</font></span>: 
<font color="Maroon" title="c47">Ss</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1_5_2_13"><span class="lab"><font color="Green" title="E99">A177</font></span></a></span>;<br/>
<a NAME="E20:73_1_5_2_13"/>
( <font color="Maroon" title="c47">Ss</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c41">S</font> &amp; <font color="Maroon" title="c41">S</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span>  )
 
<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="E21:73_1_5_2_13"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c47">Ss</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E19:73_1_5_2_13"><span class="lab"><font color="Green" title="E101">A179</font></span></a></span>;<br/>
<a NAME="E22:73_1_5_2_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A180</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E23:73_1_5_2_13"/>
for <font color="Olive" title="b1">W</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> holds <br/><font color="Maroon" title="c10">Aa</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E4:73_1_5_2_13"><span class="lab"><font color="Green" title="E93">A171</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E24:73_1_5_2_13"/><span class="kw">then </span>
not <font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">YA</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E25:73_1_5_2_13"/>
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_13"><span class="lab"><font color="Green" title="E92">A170</font></span></a>, <a class="txt" href="#E22:73_1_5_2_13"><span class="lab"><font color="Green" title="E102">A180</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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="E39:73_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A181</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">XX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1_5_2"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E40:73_1_5_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c40">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_14"/><span class="lab"><font color="Green" title="E93">A182</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_14"/><span class="lab"><font color="Green" title="E94">A183</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c41">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E4:73_1_5_2_14"/><span class="lab"><font color="Green" title="E95">A184</font></span>: 
 ex <font color="Olive" title="b1">W</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> st <br/>( <font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E39:73_1_5_2"><span class="lab"><font color="Green" title="E92">A181</font></span></a>, <a class="txt" href="#E1:73_1_5_2_14"><span class="lab"><font color="Green" title="E93">A182</font></span></a></span>;<br/>
<a NAME="E5:73_1_5_2_14"/>
<font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_14"><span class="lab"><font color="Green" title="E94">A183</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</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="E7:73_1_5_2_14"/><span class="lab"><font color="Green" title="E96">A185</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">S</font>
 <span class="kw">and </span><br/><a NAME="E8:73_1_5_2_14"/>
verum
 ;<br/>
<a NAME="E9:73_1_5_2_14"/>
<font color="Maroon" title="c41">y</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_14"><span class="lab"><font color="Green" title="E94">A183</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c43">W</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="E11:73_1_5_2_14"/><span class="lab"><font color="Green" title="E97">A186</font></span>: 
<font color="Maroon" title="c41">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">W</font>
 <span class="kw">and </span><br/><a NAME="E12:73_1_5_2_14"/><span class="lab"><font color="Green" title="E98">A187</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c43">W</font>
 ;<br/>
<a NAME="E13:73_1_5_2_14"/><span class="lab"><font color="Green" title="E99">A188</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c42">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <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="#E34:73_1_5_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a>, <a class="txt" href="#E15:73_1_5_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="txt" href="#E35:73_1_5_2"><span class="lab"><font color="Green" title="E89">A97</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/>
<a NAME="E14:73_1_5_2_14"/><span class="lab"><font color="Green" title="E100">A189</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c41">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c44">u</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="c44">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c44">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c41">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c45">FS</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c42">S</font>;<br/>
<span class="kw">assume </span><a NAME="E1:73_1_5_2_14_1"/>
<font color="Maroon" title="c44">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c44">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c41">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c46">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E101">A190</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c46">s</font>,<font color="Maroon" title="c44">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E102">A191</font></span>: 
<font color="Maroon" title="c46">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/>
<a NAME="E5:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E103">A192</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c46">s</font>,<font color="Maroon" title="c44">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E101">A190</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E6:73_1_5_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c44">u</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c47">U</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="E8:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E104">A193</font></span>: 
<font color="Maroon" title="c44">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">U</font>
 <span class="kw">and </span><br/><a NAME="E9:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E105">A194</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c47">U</font>
 ;<br/>
<a NAME="E10:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E106">A195</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c42">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:73_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="simplex0.html#T17" title="SIMPLEX0:th.17">SIMPLEX0:17</a></span>;<br/>
<a NAME="E11:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E107">A196</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c42">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_14"><span class="lab"><font color="Green" title="E95">A184</font></span></a>, <a class="txt" href="#E7:73_1_5_2_14"><span class="lab"><font color="Green" title="E96">A185</font></span></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c48">FS</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c42">S</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c42">S</font>,<font color="Maroon" title="c7">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E106">A195</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<a NAME="E13:73_1_5_2_14_1"/><span class="lab"><font color="Green" title="E108">A197</font></span>: 
<font color="Maroon" title="c43">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c42">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E3:73_1_5_2_14"><span class="lab"><font color="Green" title="E94">A183</font></span></a>, <a class="txt" href="#E7:73_1_5_2_14"><span class="lab"><font color="Green" title="E96">A185</font></span></a>, <a class="txt" href="#E11:73_1_5_2_14"><span class="lab"><font color="Green" title="E97">A186</font></span></a></span>;<br/>
<a NAME="E14:73_1_5_2_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A198</font></span>: 
<font color="Maroon" title="c48">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c43">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c43">W</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="E15:73_1_5_2_14_1"/>
<font color="Maroon" title="c46">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_5_2_14"><span class="lab"><font color="Green" title="E96">A185</font></span></a>, <a class="txt" href="#E4:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E102">A191</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:73_1_5_2_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A199</font></span>: 
<font color="Maroon" title="c47">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c42">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E5:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E103">A192</font></span></a>, <a class="txt" href="#E8:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E104">A193</font></span></a></span>;<br/>
<a NAME="E17:73_1_5_2_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A200</font></span>: 
<font color="Maroon" title="c48">FS</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c47">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c47">U</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="E18:73_1_5_2_14_1"/>
<font color="Maroon" title="c48">FS</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E107">A196</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E19:73_1_5_2_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A201</font></span>: 
<font color="Maroon" title="c48">FS</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E13:73_1_5_2_14"><span class="lab"><font color="Green" title="E99">A188</font></span></a>, <a class="ref" href="stirl2_1.html#T70" title="STIRL2_1:th.70">STIRL2_1:70</a></span>;<br/>
<a NAME="E20:73_1_5_2_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A202</font></span>: 
<font color="Maroon" title="c47">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c43">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:73_1_5_2_14"><span class="lab"><font color="Green" title="E98">A187</font></span></a>, <a class="txt" href="#E9:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E105">A194</font></span></a>, <a class="txt" href="#E10:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E106">A195</font></span></a>, <a class="txt" href="#E14:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E109">A198</font></span></a>, <a class="txt" href="#E16:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E110">A199</font></span></a>, <a class="txt" href="#E17:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E111">A200</font></span></a>, <a class="ref" href="funct_1.html#T157" title="FUNCT_1:th.157">FUNCT_1:157</a></span>;<br/>
<a NAME="E21:73_1_5_2_14_1"/>
<font color="Maroon" title="c43">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c47">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:73_1_5_2_14"><span class="lab"><font color="Green" title="E98">A187</font></span></a>, <a class="txt" href="#E9:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E105">A194</font></span></a>, <a class="txt" href="#E10:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E106">A195</font></span></a>, <a class="txt" href="#E13:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E108">A197</font></span></a>, <a class="txt" href="#E14:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E109">A198</font></span></a>, <a class="txt" href="#E17:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E111">A200</font></span></a>, <a class="txt" href="#E19:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E112">A201</font></span></a>, <a class="ref" href="funct_1.html#T157" title="FUNCT_1:th.157">FUNCT_1:157</a></span>;<br/>
<a NAME="E22:73_1_5_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c44">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1_5_2_14"><span class="lab"><font color="Green" title="E97">A186</font></span></a>, <a class="txt" href="#E8:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E104">A193</font></span></a>, <a class="txt" href="#E20:73_1_5_2_14_1"><span class="lab"><font color="Green" title="E113">A202</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:73_1_5_2_14_1"/>
<font color="Maroon" title="c44">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c41">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E15:73_1_5_2_14"/>
( <font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c41">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2_14"><span class="lab"><font color="Green" title="E93">A182</font></span></a>, <a class="txt" href="#E3:73_1_5_2_14"><span class="lab"><font color="Green" title="E94">A183</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:73_1_5_2_14"/><span class="kw">then </span>
<font color="Maroon" title="c41">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/>
<a NAME="E17:73_1_5_2_14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c40">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c41">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:73_1_5_2_14"><span class="lab"><font color="Green" title="E100">A189</font></span></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>;<br/>
<a NAME="E18:73_1_5_2_14"/><span class="kw">then </span>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>,<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c41">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D16" title="RELAT_1:def.16">RELAT_1:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:73_1_5_2_14"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</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">then </span> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">R1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default">1 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1_5_2"><span class="lab"><font color="Green" title="E72">A40</font></span></a>, <a class="ref" href="simplex1.html#T1" target="_self" title="SIMPLEX1:th.1">Th1</a></span>
<br/>.= 
1 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T29" title="CARD_2:th.29">CARD_2:29</a></span>
<br/>.= 
 <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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">R1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c25">YA</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T33" title="CARD_2:th.33">CARD_2:33</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A203</font></span>:  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c28">R1</font> = 
<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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">XX</font></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2: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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:73_1_5_2"><span class="lab"><font color="Green" title="E73">A44</font></span></a>, <a class="txt" href="#E36:73_1_5_2"><span class="lab"><font color="Green" title="E90">A98</font></span></a>, <a class="txt" href="#E39:73_1_5_2"><span class="lab"><font color="Green" title="E92">A181</font></span></a>, <a class="ref" href="simplex1.html#T1" target="_self" title="SIMPLEX1:th.1">Th1</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c17">XX</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1_5_2"><span class="lab"><font color="Green" title="E74">A45</font></span></a>, <a class="ref" href="card_2.html#T51" title="CARD_2:th.51">CARD_2:51</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="c17">XX</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex1.html#E30"><span class="lab"><font color="Green" title="E25">Lm1</font></span></a></span>
;<br/><a NAME="E43:73_1_5_2"/><span class="lab"><font color="Green" title="E94">A204</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="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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">A</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> &amp; <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="c7">A</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c7">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T8" target="_self" title="SIMPLEX1:th.8">Th8</a>, <a class="ref" href="simplex1.html#T10" target="_self" title="SIMPLEX1:th.10">Th10</a></span>;<br/><a NAME="E44:73_1_5_2"/><span class="lab"><font color="Green" title="E95">A205</font></span>: 
<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c40">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="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  or <font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_17"/>
<font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</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="c40">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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="E3:73_1_5_2_17"/><span class="lab"><font color="Green" title="E96">A206</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_17"/><span class="lab"><font color="Green" title="E97">A207</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Aa</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_5_2_17"/><span class="lab"><font color="Green" title="E98">A208</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font>
 ;<br/>
<a NAME="E6:73_1_5_2_17"/><span class="lab"><font color="Green" title="E99">A209</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:73_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="simplex0.html#T17" title="SIMPLEX0:th.17">SIMPLEX0:17</a></span>;<br/>
<a NAME="E7:73_1_5_2_17"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:73_1_5_2"><span class="lab"><font color="Green" title="E94">A204</font></span></a>, <a class="ref" href="simplex1.html#T5" target="_self" title="SIMPLEX1:th.5">Th5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E9:73_1_5_2_17"/><span class="lab"><font color="Green" title="E100">A210</font></span>: 
<font color="Maroon" title="c42">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">and </span><br/><a NAME="E10:73_1_5_2_17"/><span class="lab"><font color="Green" title="E101">A211</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c42">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E5:73_1_5_2_17"><span class="lab"><font color="Green" title="E98">A208</font></span></a>, <a class="ref" href="rlaffin2.html#T23" title="RLAFFIN2:th.23">RLAFFIN2:23</a></span>;<br/>
<a NAME="E11:73_1_5_2_17"/><span class="lab"><font color="Green" title="E102">A212</font></span>: 
<font color="Maroon" title="c42">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:73_1_5_2_17"><span class="lab"><font color="Green" title="E100">A210</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c43">B</font> = <font color="Maroon" title="c42">B</font> as   <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/>
<a NAME="E13:73_1_5_2_17"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c43">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> <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="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E9:73_1_5_2_17"><span class="lab"><font color="Green" title="E100">A210</font></span></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></span>;<br/>
<a NAME="E14:73_1_5_2_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A213</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c43">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:73_1_5_2_17"/><span class="lab"><font color="Green" title="E104">A214</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c43">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c44">y</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="c44">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font> or <font color="Maroon" title="c44">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_17_1"/>
<font color="Maroon" title="c44">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</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="c44">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c45">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_17_1"/><span class="lab"><font color="Green" title="E105">A215</font></span>: 
<font color="Maroon" title="c45">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_17_1"/><span class="lab"><font color="Green" title="E106">A216</font></span>: 
<font color="Maroon" title="c45">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_5_2_17_1"/><span class="lab"><font color="Green" title="E107">A217</font></span>: 
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_17"><span class="lab"><font color="Green" title="E97">A207</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E6:73_1_5_2_17_1"/>
<font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T2" title="RLAFFIN1:th.2">RLAFFIN1:2</a></span>;<br/>
<a NAME="E7:73_1_5_2_17_1"/><span class="kw">then </span>
<font color="Maroon" title="c45">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_17_1"><span class="lab"><font color="Green" title="E106">A216</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_5_2_17_1"/>
<font color="Maroon" title="c44">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c43">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E10:73_1_5_2_17"><span class="lab"><font color="Green" title="E101">A211</font></span></a>, <a class="txt" href="#E11:73_1_5_2_17"><span class="lab"><font color="Green" title="E102">A212</font></span></a>, <a class="txt" href="#E3:73_1_5_2_17_1"><span class="lab"><font color="Green" title="E105">A215</font></span></a>, <a class="txt" href="#E5:73_1_5_2_17_1"><span class="lab"><font color="Green" title="E107">A217</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="E16:73_1_5_2_17"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">Aa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c43">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E17:73_1_5_2_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A218</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E14:73_1_5_2_17"><span class="lab"><font color="Green" title="E103">A213</font></span></a>, <a class="txt" href="#E15:73_1_5_2_17"><span class="lab"><font color="Green" title="E104">A214</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E18:73_1_5_2_17"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_17"><span class="lab"><font color="Green" title="E97">A207</font></span></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c44">FS</font> = <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c41">S</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c10">Aa</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_5_2_17"><span class="lab"><font color="Green" title="E99">A209</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<a NAME="E20:73_1_5_2_17"/><span class="lab"><font color="Green" title="E106">A219</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="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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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="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="c7">A</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="#E22:73_1"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="simplex0.html#D13" title="SIMPLEX0:def.13">SIMPLEX0:def 13</a></span>;<br/>
<a NAME="E21:73_1_5_2_17"/><span class="lab"><font color="Green" title="E107">A220</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:73_1_5_2"><span class="lab"><font color="Green" title="E88">A96</font></span></a>, <a class="txt" href="#E32:73_1_5_2"><span class="lab"><font color="Green" title="E86">A84</font></span></a>, <a class="txt" href="#E24:73_1_5_2"><span class="lab"><font color="Green" title="E81">A72</font></span></a>, <a class="txt" href="#E15:73_1_5_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="ref" href="simplex0.html#D18" title="SIMPLEX0:def.18">SIMPLEX0:def 18</a></span>;<br/>
<a NAME="E22:73_1_5_2_17"/><span class="kw">then </span>
not <font color="Maroon" title="c41">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2"><span class="lab"><font color="Green" title="E68">A30</font></span></a></span>;<br/>
<a NAME="E23:73_1_5_2_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A221</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T20" title="RLAFFIN2:th.20">RLAFFIN2:20</a></span>;<br/>
<a NAME="E24:73_1_5_2_17"/>
 <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T5" title="RLAFFIN2:th.5">RLAFFIN2:5</a></span>;<br/>
<a NAME="E25:73_1_5_2_17"/><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:73_1_5_2_17"><span class="lab"><font color="Green" title="E108">A221</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c45">w</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E27:73_1_5_2_17"/><span class="lab"><font color="Green" title="E109">A222</font></span>: 
not <font color="Maroon" title="c45">w</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">and </span><br/><a NAME="E28:73_1_5_2_17"/><span class="lab"><font color="Green" title="E110">A223</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c45">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:73_1_5_2"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="txt" href="#E10:73_1_5_2_17"><span class="lab"><font color="Green" title="E101">A211</font></span></a>, <a class="txt" href="#E17:73_1_5_2_17"><span class="lab"><font color="Green" title="E105">A218</font></span></a>, <a class="ref" href="simplex1.html#D3" target="_self" title="SIMPLEX1:def.3">Def3</a></span>;<br/>
<a NAME="E29:73_1_5_2_17"/>
<font color="Maroon" title="c45">w</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="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="c10">Aa</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="#E27:73_1_5_2_17"><span class="lab"><font color="Green" title="E109">A222</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E30:73_1_5_2_17"/><span class="kw">then </span>
<font color="Maroon" title="c45">w</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="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="c7">A</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="#E20:73_1_5_2_17"><span class="lab"><font color="Green" title="E106">A219</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c46">W</font> = <font color="Maroon" title="c45">w</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E32:73_1_5_2_17"/>
 <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c46">W</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:73_1_5_2_17"><span class="lab"><font color="Green" title="E108">A221</font></span></a>, <a class="txt" href="#E28:73_1_5_2_17"><span class="lab"><font color="Green" title="E110">A223</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E33:73_1_5_2_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A224</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c45">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T26" target="_self" title="SIMPLEX1:th.26">Th26</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c47">s</font> = <font color="Maroon" title="c41">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex1.html#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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c48">s</font> = <font color="Maroon" title="c47">s</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</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="c10">Aa</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="#E27:73_1_5_2_17"><span class="lab"><font color="Green" title="E109">A222</font></span></a>, <a class="txt" href="#E33:73_1_5_2_17"><span class="lab"><font color="Green" title="E111">A224</font></span></a>, <a class="ref" href="matroid0.html#T1" title="MATROID0:th.1">MATROID0:1</a></span>;<br/>
<a NAME="E36:73_1_5_2_17"/><span class="lab"><font color="Green" title="E112">A225</font></span>: 
<font color="Maroon" title="c37">FA</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c48">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_17"><span class="lab"><font color="Green" title="E97">A207</font></span></a>, <a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a>, <a class="ref" href="simplex0.html#T17" title="SIMPLEX0:th.17">SIMPLEX0:17</a></span>;<br/>
<a NAME="E37:73_1_5_2_17"/>
<font color="Maroon" title="c48">s</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c10">Aa</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="#E37:73_1_5_2"><span class="lab"><font color="Green" title="E91">A169</font></span></a>, <a class="txt" href="#E21:73_1_5_2_17"><span class="lab"><font color="Green" title="E107">A220</font></span></a>, <a class="ref" href="simplex0.html#T48" title="SIMPLEX0:th.48">SIMPLEX0:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:73_1_5_2_17"/>
<font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_17"><span class="lab"><font color="Green" title="E96">A206</font></span></a>, <a class="txt" href="#E36:73_1_5_2_17"><span class="lab"><font color="Green" title="E112">A225</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="E45:73_1_5_2"/><span class="lab"><font color="Green" title="E96">A226</font></span>: 
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <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="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E37:73_1_5_2"><span class="lab"><font color="Green" title="E91">A169</font></span></a>, <a class="ref" href="simplex0.html#T26" title="SIMPLEX0:th.26">SIMPLEX0:26</a></span>;<br/><a NAME="E46:73_1_5_2"/>
<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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_18"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:73_1_5_2_18"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a></span>;<br/>
<a NAME="E2:73_1_5_2_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A227</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:73_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c40">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="c40">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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span>  or <font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:73_1_5_2_18"/>
<font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</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="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c10">Aa</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="E5:73_1_5_2_18"/><span class="lab"><font color="Green" title="E98">A228</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E6:73_1_5_2_18"/><span class="lab"><font color="Green" title="E99">A229</font></span>: 
<font color="Maroon" title="c37">FA</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Aa</font>
 ;<br/>
<a NAME="E7:73_1_5_2_18"/>
<font color="Maroon" title="c6">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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#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="c10">Aa</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="#E11:73_1"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E45:73_1_5_2"><span class="lab"><font color="Green" title="E96">A226</font></span></a>, <a class="ref" href="simplex1.html#T32" target="_self" title="SIMPLEX1:th.32">Th32</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c42">S1</font> = <font color="Maroon" title="c40">x</font> as    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="#E22:73_1"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="txt" href="#E5:73_1_5_2_18"><span class="lab"><font color="Green" title="E98">A228</font></span></a>, <a class="ref" href="simplex0.html#T49" title="SIMPLEX0:th.49">SIMPLEX0:49</a></span>;<br/>
<a NAME="E9:73_1_5_2_18"/><span class="lab"><font color="Green" title="E100">A230</font></span>: 
<font color="Maroon" title="c37">FA</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">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>, <a class="ref" href="simplex0.html#T17" title="SIMPLEX0:th.17">SIMPLEX0:17</a></span>;<br/>
<a NAME="E10:73_1_5_2_18"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c10">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:73_1_5_2"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="ref" href="simplex1.html#T5" target="_self" title="SIMPLEX1:th.5">Th5</a></span>;<br/>
<a NAME="E11:73_1_5_2_18"/><span class="kw">then </span>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c42">S1</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_5_2_18"><span class="lab"><font color="Green" title="E97">A227</font></span></a>, <a class="txt" href="#E5:73_1_5_2_18"><span class="lab"><font color="Green" title="E98">A228</font></span></a>, <a class="ref" href="rlaffin2.html#T7" title="RLAFFIN2:th.7">RLAFFIN2:7</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:73_1_5_2_18"/>
<font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_18"><span class="lab"><font color="Green" title="E98">A228</font></span></a>, <a class="txt" href="#E6:73_1_5_2_18"><span class="lab"><font color="Green" title="E99">A229</font></span></a>, <a class="txt" href="#E9:73_1_5_2_18"><span class="lab"><font color="Green" title="E100">A230</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="E47:73_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A231</font></span>: 
<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 <font color="Maroon" title="c6">m</font> <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="c10">Aa</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c37">FA</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="c10">Aa</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:73_1_5_2"><span class="lab"><font color="Green" title="E95">A205</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E48:73_1_5_2"/><span class="lab"><font color="Green" title="E98">A232</font></span>: 
( <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E49:73_1_5_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">DX</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span>,<font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_19"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c40">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">DX</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_19"/><span class="lab"><font color="Green" title="E99">A233</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">DX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E2:73_1_5_2_19"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c40">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E3:73_1_5_2_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A234</font></span>: 
<font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">S</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font> <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="c7">A</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="E5:73_1_5_2_19"/><span class="lab"><font color="Green" title="E101">A235</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 <span class="kw">and </span><br/><a NAME="E6:73_1_5_2_19"/><span class="lab"><font color="Green" title="E102">A236</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c41">S</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c42">XX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> ;<br/>
<a NAME="E7:73_1_5_2_19"/>
not <font color="Maroon" title="c40">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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2_19"><span class="lab"><font color="Green" title="E99">A233</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:73_1_5_2_19"/><span class="kw">then </span>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c41">S</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_19"><span class="lab"><font color="Green" title="E101">A235</font></span></a>, <a class="txt" href="#E6:73_1_5_2_19"><span class="lab"><font color="Green" title="E102">A236</font></span></a></span>;<br/>
<a NAME="E9:73_1_5_2_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A237</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="simplex1.html#T45" target="_self" title="SIMPLEX1:th.45">Th45</a></span>;<br/>
<a NAME="E10:73_1_5_2_19"/><span class="lab"><font color="Green" title="E104">A238</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c43">w</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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_19_1"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_5_2_19_1"/><span class="lab"><font color="Green" title="E105">A239</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">s</font>,<font color="Maroon" title="c43">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_19_1"/><span class="lab"><font color="Green" title="E106">A240</font></span>: 
<font color="Maroon" title="c44">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/>
<a NAME="E5:73_1_5_2_19_1"/><span class="lab"><font color="Green" title="E107">A241</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">s</font>,<font color="Maroon" title="c43">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_19_1"><span class="lab"><font color="Green" title="E105">A239</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E6:73_1_5_2_19_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">w</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a></span>;<br/>
<a NAME="E7:73_1_5_2_19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A242</font></span>: 
 ex <font color="Olive" title="b1">W</font> being   <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> st <font color="Maroon" title="c43">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font>
 
;<br/>
<a NAME="E8:73_1_5_2_19_1"/>
<font color="Maroon" title="c44">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_5_2_19_1"><span class="lab"><font color="Green" title="E106">A240</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:73_1_5_2_19_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c43">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E5:73_1_5_2_19_1"><span class="lab"><font color="Green" title="E107">A241</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:73_1_5_2_19_1"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_5_2_19_1"><span class="lab"><font color="Green" title="E108">A242</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="E11:73_1_5_2_19"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5_2_19_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c43">w</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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  or <font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_5_2_19_2"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</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="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">W</font> being    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</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="E3:73_1_5_2_19_2"/><span class="lab"><font color="Green" title="E105">A243</font></span>: 
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">W</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_5_2_19_2"/><span class="lab"><font color="Green" title="E106">A244</font></span>: 
<font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c44">W</font>
 ;<br/>
<a NAME="E5:73_1_5_2_19_2"/>
<font color="Maroon" title="c44">W</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 <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : verum </span>}</span> 
 
;<br/>
<a NAME="E6:73_1_5_2_19_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c44">W</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1_5_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E3:73_1_5_2_19"><span class="lab"><font color="Green" title="E100">A234</font></span></a>, <a class="txt" href="#E5:73_1_5_2_19"><span class="lab"><font color="Green" title="E101">A235</font></span></a>, <a class="txt" href="#E4:73_1_5_2_19_2"><span class="lab"><font color="Green" title="E106">A244</font></span></a></span>;<br/>
<a NAME="E7:73_1_5_2_19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A245</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">S</font>,<font color="Maroon" title="c44">W</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_5_2_19"><span class="lab"><font color="Green" title="E99">A233</font></span></a>, <a class="txt" href="#E5:73_1_5_2_19"><span class="lab"><font color="Green" title="E101">A235</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E8:73_1_5_2_19_2"/>
<font color="Maroon" title="c41">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:73_1_5_2_19_2"/>
<font color="Maroon" title="c43">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_5_2_19_2"><span class="lab"><font color="Green" title="E105">A243</font></span></a>, <a class="txt" href="#E7:73_1_5_2_19_2"><span class="lab"><font color="Green" title="E107">A245</font></span></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</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="E12:73_1_5_2_19"/><span class="kw">then </span>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">S1</font> where <font color="Olive" title="b1">S1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <font color="Maroon" title="c6">m</font>, <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c41">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S1</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1_5_2_19"><span class="lab"><font color="Green" title="E104">A238</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:73_1_5_2_19"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span>,<font color="Maroon" title="c40">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_5_2_19"><span class="lab"><font color="Green" title="E101">A235</font></span></a>, <a class="txt" href="#E9:73_1_5_2_19"><span class="lab"><font color="Green" title="E103">A237</font></span></a>, <a class="ref" href="relat_1.html#D16" title="RELAT_1:def.16">RELAT_1:def 16</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">then </span> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c31">R</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default">2 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T1" target="_self" title="SIMPLEX1:th.1">Th1</a></span>
<br/>.= 
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default">2 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:73_1_5_2"><span class="lab"><font color="Green" title="E80">A68</font></span></a></span>
<br/>.= 
2 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T29" title="CARD_2:th.29">CARD_2:29</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A246</font></span>:  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c31">R</font> = 
<span class="p1">(<span class="default">2 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default">1 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p3">{<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 <font color="Maroon" title="c6">m</font> <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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : ( <font color="Maroon" title="c14">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="c10">Aa</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">A</font> ) </span>}</span> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:73_1_5_2"><span class="lab"><font color="Green" title="E87">A85</font></span></a>, <a class="ref" href="simplex1.html#T1" target="_self" title="SIMPLEX1:th.1">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:73_1_5_2"><span class="lab"><font color="Green" title="E85">A83</font></span></a>, <a class="txt" href="#E47:73_1_5_2"><span class="lab"><font color="Green" title="E97">A231</font></span></a>, <a class="ref" href="card_2.html#T33" title="CARD_2:th.33">CARD_2:33</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> 2</span>)</span> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex1.html#E30"><span class="lab"><font color="Green" title="E25">Lm1</font></span></a></span>
<br/>.= 
<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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T52" title="CARD_2:th.52">CARD_2:52</a></span>
<br/>.= 
<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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2: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"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex1.html#E30"><span class="lab"><font color="Green" title="E25">Lm1</font></span></a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:73_1_5_2"><span class="lab"><font color="Green" title="E98">A232</font></span></a>, <a class="ref" href="card_2.html#T51" title="CARD_2:th.51">CARD_2:51</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex1.html#E30"><span class="lab"><font color="Green" title="E25">Lm1</font></span></a></span>
;<br/><a NAME="E52:73_1_5_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">XX</font> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span></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="txt" href="#E18:73_1_5_2"><span class="lab"><font color="Green" title="E77">A48</font></span></a>, <a class="txt" href="#E42:73_1_5_2"><span class="lab"><font color="Green" title="E93">A203</font></span></a></span>;<br/><a NAME="E53:73_1_5_2"/><span class="kw">then </span>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c35">DX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><a NAME="E54:73_1_5_2"/><span class="kw">then </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="c35">DX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T81" title="XREAL_1:th.81">XREAL_1:81</a></span>;<br/><a NAME="E55:73_1_5_2"/><span class="kw">then </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="c35">DX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E56:73_1_5_2"/><span class="kw">then </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="c35">DX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T21" title="INT_1:th.21">INT_1:21</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">cnc</font> = <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="c35">DX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c39">n</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">YA</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c40">cnc</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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="Maroon" title="c40">cnc</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/><span class="kw">thus </span><a NAME="E58:73_1_5_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</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="c7">A</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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c14">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="c7">A</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="Maroon" title="c40">cnc</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="txt" href="#E18:73_1_5_2"><span class="lab"><font color="Green" title="E77">A48</font></span></a>, <a class="txt" href="#E42:73_1_5_2"><span class="lab"><font color="Green" title="E93">A203</font></span></a>, <a class="txt" href="#E51:73_1_5_2"><span class="lab"><font color="Green" title="E99">A246</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:73"/><span class="lab"><font color="Green" title="E52">A247</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">A</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"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies 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="c6">A</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="c6">A</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="c6">A</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="c6">A</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">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="b3">S</font> where <font color="Olive" title="b3">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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Olive" title="b1">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="Maroon" title="c6">A</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="b2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_2"/><span class="lab"><font color="Green" title="E53">A248</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</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">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="c6">A</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="c6">A</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="c6">A</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="c6">A</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">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="b3">S</font> where <font color="Olive" title="b3">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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Olive" title="b1">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="Maroon" title="c6">A</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="b2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/>

<a NAME="E2:73_2"/><span class="lab"><font color="Green" title="E54">A249</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_2"><span class="lab"><font color="Green" title="E53">A248</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E3:73_2"/><span class="lab"><font color="Green" title="E55">A250</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="c6">A</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> <font color="Maroon" title="c2">V</font> &amp;  <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="c6">A</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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font> )
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">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="c6">A</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="c6">A</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="c6">A</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="c6">A</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="c8">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">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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</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><br/>

<span class="kw">assume </span><a NAME="E4:73_2"/>
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="c6">A</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="c6">A</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="c8">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">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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</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><br/>

<span class="kw">set </span><font color="Maroon" title="c9">X</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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</font> </span>}</span> ;<br/>
<span class="kw">take </span>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/>

<a NAME="E5:73_2"/><span class="lab"><font color="Green" title="E56">A251</font></span>: 
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
;<br/>
<a NAME="E6:73_2"/><span class="lab"><font color="Green" title="E57">A252</font></span>: 
<font color="Maroon" title="c5">Z</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/>
<a NAME="E7:73_2"/><span class="kw">then </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="c6">A</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="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_2"><span class="lab"><font color="Green" title="E53">A248</font></span></a>, <a class="ref" href="simplex0.html#T26" title="SIMPLEX0:th.26">SIMPLEX0:26</a></span>;<br/>
<a NAME="E8:73_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A253</font></span>: 
 <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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">A</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="#E3:73_2"><span class="lab"><font color="Green" title="E55">A250</font></span></a>, <a class="txt" href="#E5:73_2"><span class="lab"><font color="Green" title="E56">A251</font></span></a>, <a class="ref" href="simplex1.html#T16" target="_self" title="SIMPLEX1:th.16">Th16</a>, <a class="ref" href="simplex1.html#T22" target="_self" title="SIMPLEX1:th.22">Th22</a></span>;<br/>
<a NAME="E9:73_2"/><span class="lab"><font color="Green" title="E59">A254</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="c6">A</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="c6">A</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/>
<a NAME="E10:73_2"/><span class="lab"><font color="Green" title="E60">A255</font></span>: 
<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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</font> </span>}</span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_2_1"/><span class="lab"><font color="Green" title="E61">A256</font></span>: 
<font color="Maroon" title="c10">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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">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="c6">A</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="c6">A</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="E3:73_2_1"/><span class="lab"><font color="Green" title="E62">A257</font></span>: 
<font color="Maroon" title="c11">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">and </span><br/><a NAME="E4:73_2_1"/>
<font color="Maroon" title="c8">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_2_1"><span class="lab"><font color="Green" title="E61">A256</font></span></a></span>;<br/>
<a NAME="E5:73_2_1"/>
<font color="Maroon" title="c11">S</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="c6">A</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="#E8:73_2"><span class="lab"><font color="Green" title="E58">A253</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E6:73_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_2"><span class="lab"><font color="Green" title="E54">A249</font></span></a>, <a class="txt" href="#E9:73_2"><span class="lab"><font color="Green" title="E59">A254</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:73_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_2"><span class="lab"><font color="Green" title="E54">A249</font></span></a>, <a class="txt" href="#E3:73_2_1"><span class="lab"><font color="Green" title="E62">A257</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E11:73_2"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D1" title="ZFMISC_1:def.1">ZFMISC_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">A1</font> = <font color="Maroon" title="c6">A</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</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="c6">A</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="#E9:73_2"><span class="lab"><font color="Green" title="E59">A254</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E13:73_2"/><span class="lab"><font color="Green" title="E61">A258</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c10">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_2"><span class="lab"><font color="Green" title="E54">A249</font></span></a></span>;<br/>
<a NAME="E14:73_2"/>
<font color="Maroon" title="c10">A1</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1, <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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_2"><span class="lab"><font color="Green" title="E53">A248</font></span></a>, <a class="txt" href="#E6:73_2"><span class="lab"><font color="Green" title="E57">A252</font></span></a>, <a class="ref" href="simplex0.html#T48" title="SIMPLEX0:th.48">SIMPLEX0:48</a></span>;<br/>
<a NAME="E15:73_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">A</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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_2"><span class="lab"><font color="Green" title="E53">A248</font></span></a>, <a class="txt" href="#E8:73_2"><span class="lab"><font color="Green" title="E58">A253</font></span></a>, <a class="txt" href="#E13:73_2"><span class="lab"><font color="Green" title="E61">A258</font></span></a></span>;<br/>
<a NAME="E16:73_2"/><span class="kw">then </span>
<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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_2"><span class="lab"><font color="Green" title="E60">A255</font></span></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:73_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</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="c6">A</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="c6">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> : <font color="Maroon" title="c8">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="c6">A</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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </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="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</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="E5:73"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E52">A247</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E51">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:73"/>
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">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="b3">S</font> where <font color="Olive" title="b3">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="Olive" title="b1">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="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="b2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 ; <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>
