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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span><br/>



<b>thus </b><a NAME="E1:6"/>
<span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span><br/>

<b>then 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="E3:6_1"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c3">y</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> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self" title="RELSET_2:th.1">Th1</a></i>;<br/>
<a NAME="E4:6_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="txt" href="#E3:6_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E5:6_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:6_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="relset_2.html#T1" target="_self" title="RELSET_2:th.1">Th1</a></i>;<br/>
<a NAME="E6:6_1"/>
not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 ; <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="c5">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c3">y</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="c5">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self" title="RELSET_2:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E4:6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:6_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E4:6_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="#E3:6_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 <b>by </b><i><a class="txt" href="#E5:6_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> )</span><br/>

<b>assume </b><a NAME="E2:6"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span></span><br/>

<a NAME="E3:6"/><b>then </b><i><font color="Green" title="E5">A5</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> &amp; not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>then 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="E5:6"/><i><font color="Green" title="E6">A6</font></i>: 
( <font color="Maroon" title="c3">y</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> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self" title="RELSET_2:th.1">Th1</a></i>;<br/>
<a NAME="E6:6"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="#E3:6"><i><font color="Green" title="E5">A5</font></i></a>, <a class="txt" href="#E5:6"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="relset_2.html#T1" target="_self" title="RELSET_2:th.1">Th1</a></i>;<br/>
<a NAME="E7:6"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="#E5:6"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:6"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="relset_2.html#NK1" title="RELSET_2:NK.1">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="relset_2.html#NK1" title="RELSET_2:NK.1">}_}</a></span>
 <b>by </b><i><a class="txt" href="#E5:6"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="relset_2.html#T1" target="_self" title="RELSET_2:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
