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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="simplex0.html#NM1" title="SIMPLEX0:NM.1">SimplicialComplexStr</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c1">K</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  implies  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c1">K</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c1">K</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a> </span><br/><span class="kw">assume </span><a NAME="E2:49_1"/>
 not <font color="Maroon" title="c1">K</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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:49_1"/><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="c1">K</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="c2">D</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="E5:49_1"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">D</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="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c3">S</font> = <font color="Maroon" title="c2">D</font> as   <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="#E5:49_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a></span>;<br/><a NAME="E7:49_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">v</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> ;<br/><a NAME="E9:49_1"/>
<font color="Maroon" title="c3">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="#E5:49_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/><a NAME="E10:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font> is  <a href="simplex0.html#V1" title="SIMPLEX0:attr.1">vertex-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D3" target="_self" title="SIMPLEX0:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E11:49_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="simplex0.html#D4" target="_self" title="SIMPLEX0:def.4">Def4</a></span>; <a class="txt" 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>

<span class="kw">assume </span><a NAME="E2:49"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c1">K</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c1">K</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:49"/>
 not  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c1">K</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:49"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c1">K</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">reconsider </span><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/>
<a NAME="E7:49"/>
<font color="Maroon" title="c3">x</font> is  <a href="simplex0.html#V1" title="SIMPLEX0:attr.1">vertex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="simplex0.html#D4" target="_self" title="SIMPLEX0:def.4">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E9:49"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c4">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">and </span><br/><a NAME="E10:49"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D3" target="_self" title="SIMPLEX0:def.3">Def3</a></span>;<br/>
<a NAME="E11:49"/>
<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="#E9:49"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E12:49"/><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="c1">K</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="txt" href="#E10:49"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:49"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="simplex0.html#D7" target="_self" title="SIMPLEX0:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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