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<span class="kw">set </span><font color="Maroon" title="c3">C</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c2">Y</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">Y1</font> = <font color="Maroon" title="c2">Y</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T82" onmouseover="rs('zfmisc_1/T82')" onmouseout="rh()">ZFMISC_1:82</a></span>;<br/>
<a NAME="E2:62_1_1"/>
 <a href="simplex0.html#K2" title="SIMPLEX0:func.2">subset-closed_closure_of</a> <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
;<br/>
<a NAME="E3:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A1</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 <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T77" onmouseover="rs('zfmisc_1/T77')" onmouseout="rh()">ZFMISC_1:77</a></span>;<br/>
<a NAME="E4:62_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T81" onmouseover="rs('zfmisc_1/T81')" onmouseout="rh()">ZFMISC_1:81</a></span>;<br/>
<a NAME="E5:62_1_1"/><span class="kw">then </span>
 <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="simplex0.html#E29"><span class="lab"><font color="Green" title="E21">Lm5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:62_1_1"/>
 <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c2">Y</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</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"> verum</span><br/>


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