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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">K</font> being   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>  st  <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="c1">i</font> &amp; <font color="Maroon" title="c1">i</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">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 <font color="Olive" title="b1">K</font> st  <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="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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"> (  <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="c1">i</font> &amp; <font color="Maroon" title="c1">i</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 <font color="Maroon" title="c2">K</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110"/><span class="lab"><font color="Green" title="E42">A1</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="c1">i</font>
 <span class="kw">and </span><br/><a NAME="E2:110"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c1">i</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 <font color="Maroon" title="c2">K</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/>

<a NAME="E3:110"/>
<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1</span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</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="#E1:110"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">i1</font> = <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_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="E1:110_1"/>
(  not <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  or <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="110_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 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">  ex <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  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_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="E1:110_1_1_1"/>
(  not <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#V3" title="MATROID0:attr.3">finite-membered</a>  or  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">A</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="c2">K</font> st <br/>( <font color="Olive" title="b2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1"/>
 not <font color="Maroon" title="c2">K</font> is  <a href="matroid0.html#V3" title="MATROID0:attr.3">finite-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">  ex <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  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><div class="add"><a NAME="E2:110_1_1_1_1"/><span class="kw">then </span>
 not  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c2">K</font> is  <a href="finset_1.html#V5" title="FINSET_1:attr.5">finite-membered</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">A</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:110_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c4">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">and </span><br/><a NAME="E5:110_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 ;<br/><a NAME="E6:110_1_1_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">i1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:110_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
( <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> )
 <span class="kw">and </span><br/><a NAME="E9:110_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T10" onmouseover="rs('card_1/T10')" onmouseout="rh()">CARD_1:10</a></span>;<br/><a NAME="E10:110_1_1_1_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</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="#E4:110_1_1_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E9:110_1_1_1_1"><span class="lab"><font color="Green" title="E48">A7</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">R</font> =  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</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="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">R</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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><a NAME="E12:110_1_1_1_1"/>
<font color="Maroon" title="c6">R</font>,<font color="Maroon" title="c3">i1</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E13:110_1_1_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <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="card_1.html#D2" onmouseover="rs('card_1/D2')" onmouseout="rh()">CARD_1:def 2</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="110_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">A</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="c2">K</font> st <br/>( <font color="Olive" title="b2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</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 <font color="Maroon" title="c2">K</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">A</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="c2">K</font><span class="kw"> such that </span><br/><a NAME="E3:110_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">and </span><br/><a NAME="E4:110_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">i1</font>
 ;<br/><a NAME="E5:110_1_1_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c3">i1</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1_2"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="nat_1.html#T41" onmouseover="rs('nat_1/T41')" onmouseout="rh()">NAT_1:41</a></span>;<br/><a NAME="E6:110_1_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">i1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T3" onmouseover="rs('card_1/T3')" onmouseout="rh()">CARD_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:110_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
( <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> )
 <span class="kw">and </span><br/><a NAME="E9:110_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T10" onmouseover="rs('card_1/T10')" onmouseout="rh()">CARD_1:10</a></span>;<br/><a NAME="E10:110_1_1_1_2"/>
<font color="Maroon" title="c4">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="#E3:110_1_1_1_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a></span>;<br/><a NAME="E11:110_1_1_1_2"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</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="#E9:110_1_1_1_2"><span class="lab"><font color="Green" title="E48">A11</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">R</font> =  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</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="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">R</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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><a NAME="E13:110_1_1_1_2"/>
<font color="Maroon" title="c6">R</font>,<font color="Maroon" title="c3">i1</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_2"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E14:110_1_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <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="card_1.html#D2" onmouseover="rs('card_1/D2')" onmouseout="rh()">CARD_1:def 2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_2"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<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">  ex <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  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">d</font> =  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">K</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><span class="kw">consider </span><font color="Maroon" title="c5">S</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="E4:110_1_2"/><span class="lab"><font color="Green" title="E45">A13</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">and </span><br/><a NAME="E5:110_1_2"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">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="txt" href="#E1:110_1_2"><span class="lab"><font color="Green" title="E44">A12</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="c6">s</font> = <font color="Maroon" title="c5">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="c2">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E4:110_1_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a></span>;<br/><a NAME="E7:110_1_2"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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="#E2:110"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:110_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c3">i1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="nat_1.html#T39" onmouseover="rs('nat_1/T39')" onmouseout="rh()">NAT_1:39</a></span>;<br/><a NAME="E9:110_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">i1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">s</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E11:110_1_2"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
( <font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> )
 <span class="kw">and </span><br/><a NAME="E12:110_1_2"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T10" onmouseover="rs('card_1/T10')" onmouseout="rh()">CARD_1:10</a></span>;<br/><a NAME="E13:110_1_2"/>
<font color="Maroon" title="c5">S</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="#E4:110_1_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a></span>;<br/><a NAME="E14:110_1_2"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</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="#E12:110_1_2"><span class="lab"><font color="Green" title="E48">A16</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">R</font> =  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</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="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">R</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">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span><br/><a NAME="E16:110_1_2"/>
<font color="Maroon" title="c8">R</font>,<font color="Maroon" title="c3">i1</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:110_1_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E17:110_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <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="card_1.html#D2" onmouseover="rs('card_1/D2')" onmouseout="rh()">CARD_1:def 2</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>
