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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" 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#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" 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"> ( <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="E1:45_1"/><span class="lab"><font color="Green" title="E19">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" 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">  not <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</a> </span><br/><span class="kw">assume </span><a NAME="E2:45_1"/>
<font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</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"> contradiction</span><br/><a NAME="E3:45_1"/><span class="kw">then </span>
 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#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a> 
 ;<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#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E5:45_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R2" title="TARSKI: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>
 ;<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="#E5:45_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>;<br/><a NAME="E7:45_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="tarski.html#R2" title="TARSKI: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="E9:45_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="#E5:45_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>;<br/><a NAME="E10:45_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> 
 ;<br/><span class="kw">hence </span><a NAME="E11:45_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="simplex0.html#D4" target="_self" onmouseover="rs('simplex0/D4')" onmouseout="rh()">Def4</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>

<span class="kw">assume </span><a NAME="E2:45"/><span class="lab"><font color="Green" title="E19">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" 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#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="E3:45"/>
 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" 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"> 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">object</a> <span class="kw"> such that </span><br/><a NAME="E5:45"/><span class="lab"><font color="Green" title="E20">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>
 ;<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="#E5:45"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>;<br/>
<a NAME="E7:45"/>
<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="#E5:45"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="simplex0.html#D4" target="_self" onmouseover="rs('simplex0/D4')" onmouseout="rh()">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="E9:45"/><span class="lab"><font color="Green" title="E21">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="E10:45"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/>
<a NAME="E11:45"/>
<font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI: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="#E9:45"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>;<br/>
<a NAME="E12:45"/><span class="kw">then </span>
 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#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:45"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E19">A3</font></span></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/>


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