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<b>set </b><font color="Maroon" title="c1">X</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>assume </b><a NAME="E1:45"/>
 <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c2">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:45"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bspace.html#K9" title="BSPACE:func.9">singletons</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">g</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <b> such that </b><br/><a NAME="E5:45"/>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E6:45"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">g</font> is <a href="bspace.html#V1" title="BSPACE:attr.1">Singleton</a>
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:45"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E9:45"/>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E6:45"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:45"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:45"><i><font color="Green" title="E35">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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