<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E17">A1</font></i>: 
not <font color="Maroon" title="c1">A</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/><a NAME="E2:30_1"/><b>then </b>
not <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="realset1.html#D4" target="_self" title="REALSET1:def.4">Def4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c2">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:30_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:30_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="realset1.html#E1:30_1"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:30_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E8:30_1"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="txt" href="realset1.html#E4:30_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T41" title="ZFMISC_1:th.41">ZFMISC_1:41</a></i>;<br/><b>take </b><font color="Maroon" title="c4">a1</font> = <font color="Maroon" title="c2">a1</font>;<br/><b>take </b><font color="Maroon" title="c5">a2</font> = <font color="Maroon" title="c3">a2</font>;<br/><b>thus </b><a NAME="E9:30_1"/>
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="realset1.html#E4:30_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="realset1.html#E7:30_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="realset1.html#E8:30_1"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:30"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/>

<b>thus </b><a NAME="E4:30"/>
not <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="realset1.html#E2:30"><i><font color="Green" title="E17">A5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="realset1.html#D4" target="_self" title="REALSET1:def.4">REALSET1:def 4</a><br/>

<b>given </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E5:30"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c1">A</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>
 ;<br/>

<a NAME="E6:30"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="realset1.html#E2:30"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="realset1.html#E5:30"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E7:30"/><b>then </b>
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T26" title="ZFMISC_1:th.26">ZFMISC_1:26</a></i>;<br/>
<b>hence </b><a NAME="E8:30"/>
contradiction
 <b>by </b><i><a class="txt" href="realset1.html#E3:30"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>


</div>
