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<span class="kw">let </span><font color="Maroon" title="c1">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="simplex0.html#NV10" title="SIMPLEX0:NV.10">with_empty_element</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c1">V</font>  st <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b1">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">K</font> holds <br/> for <font color="Olive" title="b2">B</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  st (  for <font color="Olive" title="b3">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Olive" title="b1">K</font>  st  not <font color="Olive" title="b3">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><font color="Olive" title="b2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">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="Olive" title="b3">S</font></span>)</span> ) holds <br/> <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Olive" title="b2">B</font>,<font color="Olive" title="b1">K</font>) is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Olive" title="b1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="simplex0.html#NV10" title="SIMPLEX0:NV.10">with_empty_element</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">K</font> implies  for <font color="Olive" title="b1">B</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  st (  for <font color="Olive" title="b2">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>  st  not <font color="Olive" title="b2">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><font color="Olive" title="b1">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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="Olive" title="b2">S</font></span>)</span> ) holds <br/> <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c2">K</font>) is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  st (  for <font color="Olive" title="b2">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>  st  not <font color="Olive" title="b2">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><font color="Olive" title="b1">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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="Olive" title="b2">S</font></span>)</span> ) holds <br/> <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c2">K</font>) is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>  st  not <font color="Olive" title="b1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b1">S</font></span>)</span> ) implies  <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>) is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:23"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 for <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>  st  not <font color="Olive" title="b1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b1">S</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>) is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">P</font> =  <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>);<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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">A</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">x</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="Olive" title="b2">A</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b3">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b4">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">S</font> &amp; <font color="Olive" title="b1">x</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="Olive" title="b4">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">A</font> );<br/>
<a NAME="E3:23"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</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="E4:23"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 for <font color="Olive" title="b1">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</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="c6">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c6">x</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="Olive" title="b1">A</font></span>)</span> &amp;  <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> <font color="Maroon" title="c5">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 holds <br/> ex <font color="Olive" title="b2">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b3">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b3">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">S</font> &amp; <font color="Maroon" title="c6">x</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="Olive" title="b3">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">A</font> be   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</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="c7">A</font></span>)</span> &amp;  <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="c5">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 implies  ex <font color="Olive" title="b1">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b2">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c6">x</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="Olive" title="b2">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c6">x</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="c7">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:23_1"/><span class="lab"><font color="Green" title="E22">A7</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="c5">n</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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b2">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c6">x</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="Olive" title="b2">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">A1</font> =  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c7">A</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/>
<a NAME="E5:23_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</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> <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="zfmisc_1.html#T25" title="ZFMISC_1:th.25">ZFMISC_1:25</a></span>;<br/>
<a NAME="E6:23_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
 for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">P</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> holds <br/> not <font color="Olive" title="b1">P</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<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> is  <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<a NAME="E8:23_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A1</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="c7">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">BA</font> = <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E10:23_1"/>
<font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 
;<br/>
<a NAME="E11:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c7">A</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="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/>
<a NAME="E12:23_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_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="c7">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="relat_1.html#K11" title="RELAT_1:func.11">Im</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">A</font>)
 
<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/>
<a NAME="E13:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_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="c7">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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">BA</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="#E11:23_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="funct_1.html#T59" title="FUNCT_1:th.59">FUNCT_1:59</a></span>;<br/>
<a NAME="E14:23_1"/>
( <font color="Maroon" title="c9">BA</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="c8">A1</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c8">A1</font> <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"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="simplex1.html#T5" target="_self" title="SIMPLEX1:th.5">Th5</a></span>;<br/>
<a NAME="E15:23_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">BA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
 
;<br/>
<a NAME="E16:23_1"/><span class="kw">then </span>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</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">K</font> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">BA</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">BY</font> = <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_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="c7">A</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E12:23_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="funct_1.html#T59" title="FUNCT_1:th.59">FUNCT_1:59</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_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="E1:23_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b2">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c6">x</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="Olive" title="b2">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x1</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="c12">x2</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> implies <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c11">x1</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">and </span><br/><a NAME="E2:23_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c12">x2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><a NAME="E3:23_1_1_1_1"/>
<font color="Maroon" title="c11">x1</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="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E31">A16</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="E4:23_1_1_1_1"/>
<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1_1_1"><span class="lab"><font color="Green" title="E32">A17</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="firebrick">::  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="c11">Y</font> = <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> as   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">Y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c6">x</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="Olive" title="b1">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c11">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c10">BY</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">BY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c6">x</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="c10">BY</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c11">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><a NAME="E4:23_1_1_1"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">BA</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"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">BA</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="rlaffin1.html#T1" title="RLAFFIN1:th.1">RLAFFIN1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:23_1_1_1"/>
( <font color="Maroon" title="c10">BY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c6">x</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="c10">BY</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c11">Y</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="#E13:23_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E1:23_1_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T25" title="ZFMISC_1:th.25">ZFMISC_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="23_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b2">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c6">x</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="Olive" title="b2">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c13">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:23_1_1_2"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1_1_2"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c12">w</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"><font color="Maroon" title="c8">A1</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="c11">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:23_1_1_2"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
(  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r</font> &amp; <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">BA</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E8:23_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="rlaffin2.html#T26" title="RLAFFIN2:th.26">RLAFFIN2:26</a></span>;<br/><a NAME="E6:23_1_1_2"/>
(  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</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="c11">p</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="c8">A1</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="c11">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <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="c11">p</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="c5">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E3:23_1_1_2"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="stirl2_1.html#T55" title="STIRL2_1:th.55">STIRL2_1:55</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c15">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:23_1_1_2"/><span class="lab"><font color="Green" title="E33">A21</font></span>: 
<font color="Maroon" title="c15">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">S</font>
 <span class="kw">and </span><br/><a NAME="E9:23_1_1_2"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c12">w</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="c15">BS</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:23_1_1_2"/><span class="lab"><font color="Green" title="E35">A23</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c14">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="c11">p</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="#E1:23_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:23_1_1_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">S1</font> = <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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/><a NAME="E11:23_1_1_2"/><span class="lab"><font color="Green" title="E36">A24</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="c11">p</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="E12:23_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A25</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c14">S</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="#E10:23_1_1_2"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E13:23_1_1_2"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> holds <br/><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x1</font>, <font color="Maroon" title="c18">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> implies <font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A26</font></span>: 
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_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="E1:23_1_1_2_1_1"/>
( ( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> ) or ( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c18">x2</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> ) or ( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c17">x1</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> ) or ( <font color="Maroon" title="c17">x1</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="c18">x2</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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2_1"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_1_1"/>
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:23_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="23_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_1_2"/>
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c18">x2</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><div class="add"><a NAME="E2:23_1_1_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c17">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c18">x2</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E3:23_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23_1_1_2"><span class="lab"><font color="Green" title="E37">A25</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="E4:23_1_1_2_1_1_2"/>
<font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="23_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_1_3"/>
( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c17">x1</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><div class="add"><a NAME="E2:23_1_1_2_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c17">x1</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E3:23_1_1_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23_1_1_2"><span class="lab"><font color="Green" title="E37">A25</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="E4:23_1_1_2_1_1_3"/>
<font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="23_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E39">A27</font></span>: 
( <font color="Maroon" title="c17">x1</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="c18">x2</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><div class="add"><a NAME="E2:23_1_1_2_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">x1</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_1_2_1_1_4"/>
<font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E39">A27</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="firebrick">::  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><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">S1</font> = <font color="Maroon" title="c14">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> as   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a>, <a class="ref" href="tops_2.html#T13" title="TOPS_2:th.13">TOPS_2:13</a></span>;<br/><a NAME="E15:23_1_1_2"/><span class="lab"><font color="Green" title="E38">A28</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">BS</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">BY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1_2"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="relat_1.html#T120" title="RELAT_1:th.120">RELAT_1:120</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">BS1</font> = <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">S1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> ;<br/><a NAME="E17:23_1_1_2"/><span class="lab"><font color="Green" title="E39">A29</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="c15">BS</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="c18">BS1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23_1_1_2"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="rltopsp1.html#T20" title="RLTOPSP1:th.20">RLTOPSP1:20</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E18:23_1_1_2"/>
<font color="Maroon" title="c9">BA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">BY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:23_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E19:23_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<font color="Maroon" title="c9">BA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c18">BS1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23_1_1_2"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c17">S1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">S1</font> &amp; <font color="Maroon" title="c6">x</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="Olive" title="b1">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c18">BS1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">BS1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">S1</font> &amp; <font color="Maroon" title="c6">x</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="c18">BS1</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )</span><br/><a NAME="E20:23_1_1_2"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
 <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c18">BS1</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="c18">BS1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T41" title="CONVEX1:th.41">CONVEX1:41</a></span>;<br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">S1</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c14">S</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T78" title="ZFMISC_1:th.78">ZFMISC_1:78</a></span>
<br/>.= 
<font color="Maroon" title="c7">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_1_1_2"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E11:23_1_1_2"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/><span class="kw">hence </span><a NAME="E22:23_1_1_2"/>
( <font color="Maroon" title="c18">BS1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c17">S1</font> &amp; <font color="Maroon" title="c6">x</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="c18">BS1</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">S1</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="#E5:23_1_1_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E9:23_1_1_2"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E17:23_1_1_2"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="txt" href="#E19:23_1_1_2"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E20:23_1_1_2"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="rltopsp1.html#D1" title="RLTOPSP1:def.1">RLTOPSP1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E5:23"/><span class="lab"><font color="Green" title="E20">A32</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="23_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c5">x</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="Olive" title="b1">A</font></span>)</span> &amp;  <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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  holds <br/> ex <font color="Olive" title="b2">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b3">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b3">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">S</font> &amp; <font color="Maroon" title="c5">x</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="Olive" title="b3">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</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="c6">A</font></span>)</span> &amp;  <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  ex <font color="Olive" title="b1">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b2">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c5">x</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="Olive" title="b2">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_2"/><span class="lab"><font color="Green" title="E21">A33</font></span>: 
<font color="Maroon" title="c5">x</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="c6">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:23_2"/><span class="lab"><font color="Green" title="E22">A34</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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b2">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c5">x</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="Olive" title="b2">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )</span><br/>

<a NAME="E3:23_2"/>
 not  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c6">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="#E1:23_2"><span class="lab"><font color="Green" title="E21">A33</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_2"/>
 ex <font color="Olive" title="b1">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font> ex <font color="Olive" title="b2">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c5">x</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="Olive" title="b2">BS</font></span>)</span> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_2"><span class="lab"><font color="Green" title="E22">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:23"/><span class="lab"><font color="Green" title="E21">A35</font></span>: 
 for <font color="Olive" title="b1">n</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">n</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="#E5:23"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E19">A4</font></span></a>);<br/></span>
<span class="kw">thus </span><a NAME="E7:23"/>
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="simplex1.html#D4" target="_self" title="SIMPLEX1:def.4">SIMPLEX1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>  st  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  holds <br/> ex <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">K</font> st <br/>(  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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">A</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="Olive" title="b2">B</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E3:23_3"/><span class="lab"><font color="Green" title="E22">A36</font></span>: 
 not <font color="Maroon" title="c6">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">and </span><br/><a NAME="E4:23_3"/><span class="lab"><font color="Green" title="E23">A37</font></span>: 
<font color="Maroon" title="c5">x</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="c6">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#D3" target="_self" title="SIMPLEX1:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_3"><span class="lab"><font color="Green" title="E22">A36</font></span></a></span>;<br/>
<a NAME="E6:23_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E21">A35</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c9">BS</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:23_3"/><span class="lab"><font color="Green" title="E24">A38</font></span>: 
<font color="Maroon" title="c9">BS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">S</font>
 <span class="kw">and </span><br/><a NAME="E9:23_3"/><span class="lab"><font color="Green" title="E25">A39</font></span>: 
<font color="Maroon" title="c5">x</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="c9">BS</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:23_3"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">S</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="#E4:23_3"><span class="lab"><font color="Green" title="E23">A37</font></span></a></span>;<br/>
<a NAME="E11:23_3"/>
 not <font color="Maroon" title="c9">BS</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_3"><span class="lab"><font color="Green" title="E24">A38</font></span></a>, <a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a></span>;<br/>
<a NAME="E12:23_3"/><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="c9">BS</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="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#T5" target="_self" title="SIMPLEX1:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:23_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</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="#E9:23_3"><span class="lab"><font color="Green" title="E25">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E8:23"/>
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>  st  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  holds <br/> ex <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">K</font> st <br/>(  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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">A</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="Olive" title="b2">B</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  implies  ex <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">K</font> st <br/>(  not <font color="Olive" title="b1">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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="Olive" title="b1">B</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_4"/>
 not <font color="Maroon" title="c5">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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">K</font> st <br/>(  not <font color="Olive" title="b1">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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="Olive" title="b1">B</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E3:23_4"/><span class="lab"><font color="Green" title="E22">A40</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D20" title="SIMPLEX0:def.20">SIMPLEX0:def 20</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_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="E1:23_4_1"/>
( <font color="Maroon" title="c6">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c6">S</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="23_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_4_1_1"/><span class="lab"><font color="Green" title="E23">A41</font></span>: 
<font color="Maroon" title="c6">S</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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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">K</font> st <br/>(  not <font color="Olive" title="b1">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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="Olive" title="b1">B</font></span>)</span> )</span><br/><div class="add"><span class="kw">take </span>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">K</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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> <span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E2:23_4_1_1"/>
(  not  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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> <span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_4"><span class="lab"><font color="Green" title="E22">A40</font></span></a>, <a class="txt" href="#E1:23_4_1_1"><span class="lab"><font color="Green" title="E23">A41</font></span></a>, <a class="ref" href="relat_1.html#T116" title="RELAT_1:th.116">RELAT_1:116</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="23_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_4_1_2"/><span class="lab"><font color="Green" title="E23">A42</font></span>: 
 not <font color="Maroon" title="c6">S</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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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">K</font> st <br/>(  not <font color="Olive" title="b1">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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="Olive" title="b1">B</font></span>)</span> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c7">U</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">U</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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="c7">U</font></span>)</span> )</span><br/><a NAME="E2:23_4_1_2"/><span class="lab"><font color="Green" title="E24">A43</font></span>: 
<font color="Maroon" title="c5">A</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="c7">U</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> or <font color="Maroon" title="c8">x</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="c7">U</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_4_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</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="c7">U</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_4_1_2_1"/><span class="lab"><font color="Green" title="E25">A44</font></span>: 
<font color="Maroon" title="c9">y</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="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E4:23_4_1_2_1"/><span class="lab"><font color="Green" title="E26">A45</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">and </span><br/><a NAME="E5:23_4_1_2_1"/><span class="lab"><font color="Green" title="E27">A46</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_4"><span class="lab"><font color="Green" title="E22">A40</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c2">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_4_1_2_1"><span class="lab"><font color="Green" title="E26">A45</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<a NAME="E7:23_4_1_2_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E3:23_4_1_2_1"><span class="lab"><font color="Green" title="E25">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/>
<a NAME="E8:23_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A47</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</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="c10">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E9:23_4_1_2_1"/>
 <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="c10">y</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="c7">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_4_1_2_1"><span class="lab"><font color="Green" title="E26">A45</font></span></a>, <a class="ref" href="rltopsp1.html#T20" title="RLTOPSP1:th.20">RLTOPSP1:20</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:23_4_1_2_1"/>
<font color="Maroon" title="c8">x</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="c7">U</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_4_1_2_1"><span class="lab"><font color="Green" title="E27">A46</font></span></a>, <a class="txt" href="#E8:23_4_1_2_1"><span class="lab"><font color="Green" title="E28">A47</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:23_4_1_2"/>
<font color="Maroon" title="c7">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_4_1_2"><span class="lab"><font color="Green" title="E23">A42</font></span></a>, <a class="ref" href="simplex0.html#T9" title="SIMPLEX0:th.9">SIMPLEX0:9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:23_4_1_2"/>
(  not <font color="Maroon" title="c7">U</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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="Maroon" title="c5">A</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="c7">U</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_4_1_2"><span class="lab"><font color="Green" title="E24">A43</font></span></a>, <a class="ref" href="convex1.html#T30" title="CONVEX1:th.30">CONVEX1:30</a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
<span class="kw">hence </span><a NAME="E9:23"/>
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>  st  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  holds <br/> ex <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">K</font> st <br/>(  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &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">A</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="Olive" title="b2">B</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
