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<span class="kw">assume </span><a NAME="E1:11_1_1"/>
 not  <a href="simplex0.html#K1" title="SIMPLEX0:func.1">subset-closed_closure_of</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 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="c1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:11_1_1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#K1" title="SIMPLEX0:func.1">subset-closed_closure_of</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">x</font> = <font color="Maroon" title="c1">x</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E5:11_1_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="simplex0.html#T2" target="_self" onmouseover="rs('simplex0/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:11_1_1"/>
contradiction
 ; <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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