<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/> for <font color="Olive" title="b2">KX</font> being   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">KX</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  &amp;  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/> <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b2">KX</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">KX</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b2">KX</font> #)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">KX</font> being   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">KX</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  &amp;  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> holds <br/> <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b1">KX</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">KX</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">KX</font> #)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">KX</font> be    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">KX</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  &amp;  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> implies  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">KX</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> #) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:115"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c3">KX</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> 
 <span class="kw">and </span><br/><a NAME="E2:115"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">KX</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> #)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font>;<br/>
<a NAME="E3:115"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:115"/><span class="kw">then </span>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">KX</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<a NAME="E5:115"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
( <font color="Maroon" title="c3">KX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  or <font color="Maroon" title="c3">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</a></span>;<br/>
<a NAME="E6:115"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A4</font></span>: 
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c3">KX</font> is  <a href="matroid0.html#NV6" title="MATROID0:NV.6">finite-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#D6" title="MATROID0:def.6">MATROID0:def 6</a></span>;<br/>
<a NAME="E7:115"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> or <font color="Maroon" title="c5">x</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 <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font></span>)</span> )</span><br/>

<a NAME="E1:115_1"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">KX</font></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="c2">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="#E2:115"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:115_1"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c5">x</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="c3">KX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c5">x</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="c3">KX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:115"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="matroid0.html#D5" title="MATROID0:def.5">MATROID0:def 5</a></span>;<br/>
<a NAME="E4:115_1"/>
( <font color="Maroon" title="c6">A</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  &amp; not <font color="Maroon" title="c3">KX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="pencil_1.html#D4" title="PENCIL_1:def.4">PENCIL_1:def 4</a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E5:115_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">A</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="c3">KX</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="#E5:115"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</a></span>;<br/>
<a NAME="E6:115_1"/><span class="kw">then </span>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:115_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T41" title="NAT_1:th.41">NAT_1:41</a></span>;<br/>
<a NAME="E8:115_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="simplex0.html#D2" target="_self" title="SIMPLEX0:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:115_1"/>
<font color="Maroon" title="c5">x</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 <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T2" target="_self" title="SIMPLEX0:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:115"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
 <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> or <font color="Maroon" title="c5">x</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="c3">KX</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:115_2"/>
(  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> or <font color="Maroon" title="c5">x</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="c3">KX</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D2" target="_self" title="SIMPLEX0:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:115"/>
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c3">KX</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E44">A1</font></span></a></span>;<br/>
<a NAME="E10:115"/><span class="kw">then </span>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:115"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="simplex0.html#T3" target="_self" title="SIMPLEX0:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:115"/>
 <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">KX</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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