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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">SC</font> being   <a href="simplex0.html#NM4" title="SIMPLEX0:NM.4">SimplicialComplex</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">SC</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> holds <br/><font color="Olive" title="b1">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Olive" title="b2">A</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b1">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Olive" title="b3">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">SC</font> be   <a href="simplex0.html#NM4" title="SIMPLEX0:NM.4">SimplicialComplex</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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">SC</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Olive" title="b1">A</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">SC</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="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c3">A</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c4">B</font> )</span><br/>





<a NAME="E1:97"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">SC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">SC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T37" target="_self" onmouseover="rs('simplex0/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:97"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</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="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c3">A</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c4">B</font></span><br/>

<a NAME="E3:97"/><span class="kw">then </span>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c3">A</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> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T67" onmouseover="rs('zfmisc_1/T67')" onmouseout="rh()">ZFMISC_1:67</a></span>;<br/>
<a NAME="E4:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">SC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">SC</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/>
<a NAME="E5:97"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D16" target="_self" onmouseover="rs('simplex0/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:97"/>
<font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c3">A</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">SC</font> <a href="simplex0.html#K7" title="SIMPLEX0:func.7">|</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="simplex0.html#D13" target="_self" onmouseover="rs('simplex0/D13')" onmouseout="rh()">Def13</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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