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<b>consider </b><font color="Maroon" title="c1">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<b>consider </b><font color="Maroon" title="c2">B</font> being  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<b>consider </b><font color="Maroon" title="c3">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:12_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>
<b>take </b>
<font color="Maroon" title="c3">C</font>
;<br/>

<a NAME="E5:12_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="taxonom2.html#V3" title="TAXONOM2:attr.3">hierarchic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">TAXONOM2:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="taxonom2.html#E1:12_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_1_1_1_1"/>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="taxonom2.html#E1:12_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:12_1_1_1_1_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_2_1"/>
( <font color="Maroon" title="c3">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c3">C</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="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="taxonom2.html#E1:12_1_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_2_1_1"/>
<font color="Maroon" title="c3">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c3">C</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="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1_2_1_2"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c3">C</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">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><b>thus </b><a NAME="E4:12_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="taxonom2.html#E4:12_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="taxonom2.html#E3:12_1_1_2_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T9" title="ZFMISC_1:th.9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_1"/>
( not <font color="Maroon" title="c3">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; <font color="Maroon" title="c3">C</font> is <a href="taxonom2.html#V3" title="TAXONOM2:attr.3">hierarchic</a> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E5:12_1_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


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