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<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  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <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#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</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  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <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#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</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"> (  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <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#K6" title="SIMPLEX0:func.6">degree</a> <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">swlc</font> =  <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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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/>
<span class="kw">assume </span><a NAME="E1:113"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <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#K6" title="SIMPLEX0:func.6">degree</a> <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font></span><br/>

<a NAME="E2:113"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</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="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="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">i1</font> = <font color="Maroon" title="c2">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#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">A</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 <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">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 )</span><br/><span class="kw">assume </span><a NAME="E1:113_1"/>
<font color="Maroon" title="c7">A</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">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</span><br/><a NAME="E2:113_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</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="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:113_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
( <font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c8">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T2" target="_self" title="SIMPLEX0:th.2">Th2</a></span>;<br/><a NAME="E5:113_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">x</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> <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> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</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="c8">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:113_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="simplex0.html#D2" target="_self" title="SIMPLEX0:def.2">Def2</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/><a NAME="E6:113_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E7:113_1"/>
 <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="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A</font>
 ;<br/><span class="kw">hence </span><a NAME="E8:113_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="nat_1.html#T41" title="NAT_1:th.41">NAT_1:41</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>
<span class="kw">hence </span><a NAME="E5:113"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="simplex0.html#T25" target="_self" title="SIMPLEX0:th.25">Th25</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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