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<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><span class="kw">now </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"> (  not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp; <font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  implies  ex <font color="Olive" title="b1">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( (  for <font color="Olive" title="b2">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) &amp;  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="c1">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;  not  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) ) )</span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1;<br/><span class="kw">assume </span><a NAME="E1:67_1_1_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
(  not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp; <font color="Maroon" title="c1">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">  ex <font color="Olive" title="b1">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( (  for <font color="Olive" title="b2">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) &amp;  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="c1">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;  not  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) )</span><br/><a NAME="E2:67_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">K</font> is  <a href="finset_1.html#V5" title="FINSET_1:attr.5">finite-membered</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#D6" onmouseover="rs('matroid0/D6')" onmouseout="rh()">MATROID0:def 6</a></span>;<br/><a NAME="E3:67_1_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:67_1_1_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 <span class="kw">and </span><br/><a NAME="E6:67_1_1_1"/><span class="lab"><font color="Green" title="E29">A5</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" onmouseover="rs('nat_1/S5')" onmouseout="rh()">NAT_1:sch 5</a>(<a class="txt" href="#E3:67_1_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>);<br/></span><span class="kw">take </span><font color="Maroon" title="c3">D</font> = <font color="Maroon" title="c2">k</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</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"> ( (  for <font color="Olive" title="b1">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) &amp;  ex <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font> st <br/>( <font color="Olive" title="b1">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  &amp;  not  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) )</span><br/><span class="kw">thus </span><a NAME="E7:67_1_1_1"/>
 for <font color="Olive" title="b1">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></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">  ex <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font> st <br/>( <font color="Olive" title="b1">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  &amp;  not  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E8:67_1_1_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">D</font> <a href="xxreal_3.html#K1" title="XXREAL_3: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"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1"><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="E1:67_1_1_1_1"/>
( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:67_1_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">S</font> = <font color="Maroon" title="c4">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 <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a></span>;<br/><a NAME="E5:67_1_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c5">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a></span>;<br/><a NAME="E6:67_1_1_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:67_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_1_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E1:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E5:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></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="67_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_1_2"/>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">k1</font> = <font color="Maroon" title="c2">k</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 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="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/><a NAME="E3:67_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k1</font>]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="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="c5">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c5">S</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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font></span><br/>

<a NAME="E2:67_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</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="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/>
<a NAME="E3:67_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k1</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="#E8:67_1_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E1:67_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:67_1_1_1_1_2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font>
 <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><a NAME="E4:67_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">k1</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="c4">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:67_1_1_1_1_2"/>
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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:67_1_1"/>
( (  not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp; <font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) &amp;  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="c1">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">b<sub>1</sub></font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 ) ) ) &amp; ( <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 ) &amp; ( ( <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  or  not <font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  ) &amp;  not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</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>
