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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">KX</font> being   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">P</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">KX</font> holds <br/> ex <font color="Olive" title="b4">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">KX</font> st <br/>( <font color="Olive" title="b4">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b2">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b4">S</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">KX</font> &amp; <font color="Olive" title="b2">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b4">S</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  ) ) holds <br/> <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b2">P</font>,<font color="Olive" title="b1">KX</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">P</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> holds <br/> ex <font color="Olive" title="b3">S</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/>( <font color="Olive" title="b3">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b3">S</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b3">S</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  ) ) holds <br/> <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">P</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> holds <br/> ex <font color="Olive" title="b2">S</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/>( <font color="Olive" title="b2">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b2">S</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b2">S</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  ) ) implies  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:125"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">P</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a> 
 <span class="kw">and </span><br/><a NAME="E2:125"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> holds <br/> ex <font color="Olive" title="b2">S</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/>( <font color="Olive" title="b2">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b2">S</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Olive" title="b2">S</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">PP</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>);<br/>
<a NAME="E3:125"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="simplex0.html#T52" target="_self" onmouseover="rs('simplex0/T52')" onmouseout="rh()">Th52</a></span>;<br/>
<a NAME="E4:125"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> holds <br/> ex <font color="Olive" title="b2">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="125_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> implies  ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span><br/>

<a NAME="E1:125_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D20" target="_self" onmouseover="rs('simplex0/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:125_1"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="E4:125_1"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<font color="Maroon" title="c6">A</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">and </span><br/><a NAME="E5:125_1"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E6:125_1"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
 <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">and </span><br/><a NAME="E7:125_1"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font></span>)</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">and </span><br/><a NAME="E8:125_1"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E48">A2</font></span></a></span>;<br/>
<a NAME="E9:125_1"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:125_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E10:125_1"/>
 not <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="#E5:125_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">S</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c6">A</font><span class="kw"> such that </span><br/><a NAME="E12:125_1"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
( <font color="Maroon" title="c7">S</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  &amp; <font color="Maroon" title="c7">S</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  )
 <span class="kw">and </span><br/><a NAME="E13:125_1"/>
<font color="Maroon" title="c6">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">and </span><br/><a NAME="E14:125_1"/><span class="lab"><font color="Green" title="E59">A13</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">and </span><br/><a NAME="E15:125_1"/>
 for <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T12" target="_self" onmouseover="rs('simplex0/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E16:125_1"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T67" onmouseover="rs('zfmisc_1/T67')" onmouseout="rh()">ZFMISC_1:67</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">SS</font> = <font color="Maroon" title="c7">S</font> as   <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="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E18:125_1"/><span class="lab"><font color="Green" title="E60">A14</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="125_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:125_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font></span><br/>

<a NAME="E2:125_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:125_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:125_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#D3" onmouseover="rs('orders_1/D3')" onmouseout="rh()">ORDERS_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E19:125_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T123" onmouseover="rs('relat_1/T123')" onmouseout="rh()">RELAT_1:123</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">PS</font> = <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c8">SS</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#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E7:125_1"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E21:125_1"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a></span>;<br/>
<a NAME="E22:125_1"/>
<font color="Maroon" title="c8">SS</font> is  <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="125_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">SS</font> or <font color="Maroon" title="c10">B</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:125_1_2"/>
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">SS</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">B</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> </span><br/>

<a NAME="E2:125_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:125_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="classes1.html#D1" onmouseover="rs('classes1/D1')" onmouseout="rh()">CLASSES1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:125_1_2"/>
<font color="Maroon" title="c10">B</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E23:125_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">SS</font> is   <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="#E5:125_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E12:125_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E14:125_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">PS</font> = <font color="Maroon" title="c9">PS</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D20" target="_self" onmouseover="rs('simplex0/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<a NAME="E25:125_1"/>
<font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c6">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c7">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:125_1"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="relat_1.html#T129" onmouseover="rs('relat_1/T129')" onmouseout="rh()">RELAT_1:129</a></span>;<br/>
<a NAME="E26:125_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">PS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:125_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E8:125_1"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E9:125_1"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E14:125_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E18:125_1"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="combgras.html#T4" onmouseover="rs('combgras/T4')" onmouseout="rh()">COMBGRAS:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:125_1"/>
 ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="125_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:125_2"/>
( <font color="Maroon" title="c2">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  or ( <font color="Maroon" title="c2">K</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</a>  &amp; <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  ) or (  not <font color="Maroon" title="c2">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp;  not <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1"><span class="kw">suppose </span></a><a NAME="E1:125_2_1"/><span class="lab"><font color="Green" title="E51">A16</font></span>: 
<font color="Maroon" title="c2">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font></span><br/><div class="add"><a NAME="E2:125_2_1"/><span class="lab"><font color="Green" title="E52">A17</font></span>: 
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T23" target="_self" onmouseover="rs('simplex0/T23')" onmouseout="rh()">Th23</a></span>;<br/><a NAME="E3:125_2_1"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_2_1"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="ref" href="simplex0.html#T22" target="_self" onmouseover="rs('simplex0/T22')" onmouseout="rh()">Th22</a></span>;<br/><span class="kw">hence </span><a NAME="E4:125_2_1"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E2:125_2_1"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="125_2_2"><span class="kw">suppose </span></a><a NAME="E1:125_2_2"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
( <font color="Maroon" title="c2">K</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</a>  &amp; <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">d</font> =  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c6">dPP</font> =  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E3:125_2_2"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
 <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T23" target="_self" onmouseover="rs('simplex0/T23')" onmouseout="rh()">Th23</a></span>;<br/><a NAME="E4:125_2_2"/>
<font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_2_2"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="ref" href="simplex0.html#T22" target="_self" onmouseover="rs('simplex0/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E5:125_2_2"/><span class="kw">then </span>
 <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125_2_2"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E6:125_2_2"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T8" onmouseover="rs('int_1/T8')" onmouseout="rh()">INT_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c5">d</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E8:125_2_2"/>
 ex <font color="Olive" title="b1">S</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E50">A4</font></span></a></span>;<br/><a NAME="E9:125_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">dPP</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">d</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_2_2"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E10:125_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">dPP</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E11:125_2_2"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="125_2_3"><span class="kw">suppose </span></a><a NAME="E1:125_2_3"/>
(  not <font color="Maroon" title="c2">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp;  not <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font></span><br/><div class="add"><a NAME="E2:125_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A20</font></span>: 
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E3:125_2_3"/>
 not  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>) is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="125_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:125_2_3_1"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>) is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">d</font> = <span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p2">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:125_2_3_1"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<font color="Maroon" title="c6">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">and </span><br/><a NAME="E5:125_2_3_1"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_2_3_1"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c6">S</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125_2_3_1"><span class="lab"><font color="Green" title="E53">A22</font></span></a></span>;<br/>
<a NAME="E7:125_2_3_1"/>
 ex <font color="Olive" title="b1">S1</font> being   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E2:125_2_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T3" onmouseover="rs('xxreal_0/T3')" onmouseout="rh()">XXREAL_0:3</a></span>;<br/>
<a NAME="E8:125_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_2_3_1"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E5:125_2_3_1"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:125_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E4:125_2_3"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125_2_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
