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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  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><br/>  for <font color="Olive" title="b2">SX</font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b1">KX</font> holds   <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Olive" title="b2">SX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Olive" title="b1">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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" 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">  for <font color="Olive" title="b1">SX</font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KX</font> holds   <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Olive" title="b1">SX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c2">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">SX</font> be    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KX</font>; <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="c3">SX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c2">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c4">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="c3">SX</font> or <font color="Maroon" title="c4">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="c2">KX</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c4">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="c3">SX</font>
 ; <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="c4">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="c2">KX</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">SX</font> ;<br/>
<a NAME="E3:84"/>
<font color="Maroon" title="c5">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="txt" href="#E1:84"><span class="lab"><font color="Green" title="E33">A1</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="c6">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">SX</font><span class="kw"> such that </span><br/><a NAME="E5:84"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c6">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">and </span><br/><a NAME="E6:84"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 ;<br/>
<a NAME="E7:84"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">SX</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="c2">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D13" target="_self" onmouseover="rs('simplex0/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E8:84"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c6">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="c3">SX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/>
<a NAME="E9:84"/><span class="kw">then </span>
<font color="Maroon" title="c6">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="c2">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:84"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c6">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KX</font> ;<br/>
<a NAME="E11:84"/>
<font color="Maroon" title="c5">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c5">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">KX</font> ;<br/>
<a NAME="E13:84"/>
<font color="Maroon" title="c7">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="#E7:84"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E8:84"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>;<br/>
<a NAME="E14:84"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="txt" href="#E6:84"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:84"/>
<font color="Maroon" title="c4">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="c2">KX</font>
 <span class="kw">by</span> <span class="lab"><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/>


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