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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  <a href="simplex0.html#NM1" title="SIMPLEX0:NM.1">SimplicialComplexStr</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">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Olive" title="b1">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1,<font color="Maroon" title="c1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c1">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"> <font color="Maroon" title="c2">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1,<font color="Maroon" title="c1">K</font></span><br/>



<a NAME="E1:114"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T24" target="_self" onmouseover="rs('simplex0/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<a NAME="E2:114"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T37" onmouseover="rs('xxreal_3/T37')" onmouseout="rh()">XXREAL_3:37</a></span>;<br/>
<a NAME="E3:114"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T24" onmouseover="rs('xxreal_3/T24')" onmouseout="rh()">XXREAL_3:24</a></span>;<br/>
<a NAME="E4:114"/>
( <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1</span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 )
 
<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="E5:114"/>
<font color="Maroon" title="c2">S</font> is    <a href="simplex0.html#M3" title="SIMPLEX0:mode.3">Simplex</a> of <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1,<font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="simplex0.html#D18" target="_self" onmouseover="rs('simplex0/D18')" onmouseout="rh()">Def18</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/>


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