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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 <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#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b2">KX</font>,<font color="Olive" title="b1">i</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  &amp; not <font color="Olive" title="b2">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  holds <br/><font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</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#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b1">KX</font>,<font color="Maroon" title="c2">i</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  &amp; not <font color="Olive" title="b1">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  holds <br/><font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</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#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c3">KX</font>,<font color="Maroon" title="c2">i</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  &amp; not <font color="Maroon" title="c3">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  implies <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 )</span><br/>





<span class="kw">assume </span><a NAME="E1:111"/>
<font color="Maroon" title="c3">KX</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <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> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  or <font color="Maroon" title="c3">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  or <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 )</span><br/>

<a NAME="E2:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A1</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="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> 
 
;<br/>
<span class="kw">assume </span><a NAME="E3:111"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 <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> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a> 
 ; <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="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  or <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E4:111"/>
not <font color="Maroon" title="c3">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a> 
 ; <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="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span><br/>

<a NAME="E5:111"/><span class="kw">then </span>
not the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> is  <a href="setfam_1.html#V2" title="SETFAM_1:attr.2">empty-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D7" target="_self" title="SIMPLEX0:def.7">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:111"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c4">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="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:111"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:111"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E11:111"/><span class="kw">then </span>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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; not <font color="Maroon" title="c2">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
;<br/>
<a NAME="E12:111"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></span>;<br/>
<a NAME="E13:111"/><span class="kw">then </span>
1 <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="card_3.html#T109" title="CARD_3:th.109">CARD_3:109</a></span>;<br/>
<a NAME="E14:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>;<br/>
<a NAME="E15:111"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E16:111"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="txt" href="#E2:111"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<a NAME="E17:111"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="#E14:111"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="simplex0.html#D2" target="_self" title="SIMPLEX0:def.2">Def2</a></span>;<br/>
<a NAME="E18:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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>;<br/>
<a NAME="E19:111"/>
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> is  <a href="setfam_1.html#V2" title="SETFAM_1:attr.2">empty-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="simplex0.html#D7" target="_self" title="SIMPLEX0:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:111"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:111"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="setfam_1.html#D11" title="SETFAM_1:def.11">SETFAM_1:def 11</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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