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<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">SX</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</a> </span><br/>

<a NAME="E1:84_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c2">KX</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D5" target="_self" title="SIMPLEX0:def.5">Def5</a></span>;<br/>
<a NAME="E2:84_1_1"/>
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" title="SIMPLEX0:def.13">Def13</a></span>;<br/>
<a NAME="E3:84_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> 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>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</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="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E4:84_1_1"/>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">SX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c3">SX</font> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="simplex0.html#E29"><span class="lab"><font color="Green" title="E24">Lm5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:84_1_1"/>
<font color="Maroon" title="c3">SX</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E1:84_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="simplex0.html#D5" target="_self" title="SIMPLEX0:def.5">Def5</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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